I reviewed the post below for typos and errors, but can't say I got them all. I'll edit later if I find anything, or someone else does. These will be at the bottom of this post.

Short version: In your picture, the resistor circled in Red may be good, bad or not needed. First determine if it's needed, then you may need to remove it from the circuit to test it. You need to clean it, if possible, to accurately read the stripes in order to read the spec’s, and need to do other work to get the wattage, or use other methods. The item circled in Black is not a resistor, it’s a capacitor. The presence of the Neon Bulb is strange, and may be an unofficial modification to resolve some prior issue.

Could this explain my issue?

Probably not. The rest of this long post is how to get the resistor spec and to understand better what’s going on in your picture. You might need to do a couple of things and report the results back here. There are some suggested next steps at the end.

Long version with more explanation.

Your problem may lie elsewhere. This is little long, but since you said you are newer to troubleshooting, there’s a little more explanation. If you want to skip, see "Next Steps after the Above" at the bottom

Also in the picture is what appears to me as a burnt out resistor. It is blue so I believe that means it is a 4.7K ohm (Am I getting that right) resistor but I cannot tell what colors used to be there.

The stripes indicate the values, not the basic color. The basic color can be blue, tan or anything else. Some larger resistors have the values printed on them because they are large enough to do so. Smaller ones use stripes.

In your picture, the blue resistor circled in red is at the least a little dirty, so it’s hard to read the stripes printed on it in the photo. These stripes encode the value and tolerance. Use a gentle cleaner on a dampened qtip or cloth to try to wipe off any dirt. Once you do that, you can get the value by reading the colors of the stripes on it. You can use a site like: https://resistorcolorcodecalc.com/, or once clean, upload another photo. If you take another photo to upload, stick a piece of white paper behind it to help make the colors accurate, but first determine if you even need it. More on this later.

Black Circled Item

The item circled in black is a disc capacitor, not a resistor. The value looks to be .1 microfarads and the rating is 500V. [1] Your meter would need a capacitor measurement mode on your meter to test it. These disc capacitors are very cheap parts, if you find you need to replace it.

Basic Material

As used here below and by others: “Open” means electricity can’t pass; a turned off switch, blown fuse, a cut wire or a damaged electrical component. “Short” is the opposite, a switch turned on, an electrical component damaged in a way that it passes all electricity when it’s not supposed to, or a wire broke and fell onto another live circuit where it’s not supposed to be. “Blown” is vernacular for a damaged component, sometimes accompanied with smoke, smell or ‘mud-stains’. Smoke and Smell go away a short time later. ‘Mud-stains’ hang around, but are sometimes hidden. Sometimes a component looks ok but is broken, hence the need to test with a meter.

Test continuity, ohms and as much else as possible with the machine turned off. To check voltages [or current] the machine needs to be powered on. For safety, keep one hand in your pocket or otherwise away from the wires. An alligator clip can be used to old the black end of the probe to on ground for testing volts DC. Short will probably read on your meter as 0.00. Open probably as “OL” or “-1”, maybe “Open”.

Neon Bulb

I’m not directly familiar with GoldenEye, but the neon bulb circled in blue is strange to see here. I quickly skimmed the GoldenEye documents on IPDB and didn’t see bulbs of this type used. Maybe that document you mentioned was a later field change order. Neon bulbs need at least 60VDC to get going and for some it’s 100VDC. Higher voltages are usually used up in the backbox for the score display, not in the playfield wiring, so this is why it’s strange to see, but anything is possible.

If the blue resistor is involved with the neon bulb, it may be the ballast resistor for it [2]. Neon bulbs can be used to create constant voltage for the rest of a circuit. Voltage regulators are a better way to do this, but someone ‘clever’ may have been making use of what they had on hand to solve an issue in the past.

Do you need the resistor?

Trace the wires out of the blue resistor first both ends. If they wind up at one side of the neon bulb and the other side of the bulb is not connected, the blue resistor is not being used.
You might need to figure out where neon bulb went before it was cut, but if this is an unofficial change it could be anywhere. With the machine powered on, you might check the voltages on both sides of the blue resistor to get an idea if is getting any power at all. If voltages are present, the values might provide an idea of what circuits to look at on the diagrams. To Measure DC Volts, the black probe of the meter should go to ground, and touch the red one to both sides of the resistor.

Testing in place

With the machine powered off, if you test the blue resistor while it’s in the circuit with other components, your meter may pick up the rest of the circuit and mess with the reading. If it’s dead short, the rest of the circuit may not matter much if the shorted resistor is the path of least resistance. If it’s blown open, the same could apply with regard to the involvement of the rest of the circuit but an error is more likely. If it’s not obviously shorted or open, you should get a value that approximately matches the stripes. Resistors have tolerance and it can be up to 25% of the value so you are unlikely to get an exact value. For this type of measurement, either meter probe can be on either side of the resistor.

Isolated Testing

Since other parts of the circuit can mess up the meter reading it's best to isolate. To isolate, remove one end from the circuit by desoldering or cutting the resistor wire on one side so it is isolated from the rest of the circuit when you test it. Once tested, if it’s good and you need it, you can resolder it or a new one back in. (Other methods won’t work, you need to solder it.)

Resistor Wattage Rating

Part of a Resistor Spec is the Wattage rating. If the blue resistor is bad and is needed for the machine to operate, to replace it with the correct one, you need the correct wattage rating. This may not be printed on the resistor. If the resistor is an official part of the machine, the best place to get this from the game manual circuit diagram, links below [3]. If that information is not available in the document, a game manual for a different game from the same company and time frame may have the information. Getting down to the last resort, you might look elsewhere in your machine for a similar size and shape resistor that is documented in the circuit diagram to make a guess at the wattage. Finally, if all else fails, you might need to put a volt meter on the circuit with the machine on, find out how much voltage is present and with the value of the resistor, calculate the wattage required using the Ohm’s law formula and any circuit diagram.

The wattage of a replacement resistor has to be the same or more than what is there. If it’s 2 Watts you can use a 2W or a 4W, but can’t use a 1 Watt.

Best to get the value from the stripes first, test it, and post your results.

If you trace it out and the resistor isn’t doing anything you might not need it.

Next Steps after the Above

I'm guessing that the items in the photo are unrelated to the stated problem of no Flash lamps and no Coils, but eliminating what is not a problem, can be on the road to finding what is the problem.

Work next with with the I/O Power Driver board. See if the input voltages on J17 are correct and present. The manual in the link below has the board diagram on page 101. For output power, J6 is to the flash lamps and J7, J8 & J9 are involved with solenoids [coils]. If probing with the machine powered on, take care not to short two pins with your meter probe. Put black tape on it so only the very tip of the metal is exposed, or otherwise prevent the metal part of the probe from being able to hit two pins at once. If you hit two pins at once and short them, further damage can happen. For DC volts, black probe on ground, red to measurement. AC volts will have two feeds, both A/C and neither ground. For example, on J17, there are two BLU-WHT wires for +13VAC. Use both to measure AC voltage for those pins.

If J17 isn't providing power to the I/O Power Driver Board, the problem is between the board and main transformer, or main transformer. If power is getting in, but not out, to J7 and/or J8 and/or J9, the problem is likely within the board. If the power is making it to the output connectors and pins on the I/O Power Driver board, then the problem is downstream from there. At the least, you should have an idea where to look next.

Footnotes:

[1] Oversimplified, a Capacitor holds a little power back and uses it when needed. On older machines they may have been used across switches to make sure the MPU had time to sense the switch was closed by extending the time it looked closed. They can be used on bulbs to help prevent flickering. Sometimes they are used in older machines across flipper switches to stop arcing [sparks]. Exact values and how to do this need to be calculated and isn’t important here.
[2] Ballast resistor: LEDs and Neon bulbs need a little help from a resistor to make sure they get an appropriate amount of electrical current. Too little the bulb won’t light, too much and it will be damaged.
[3] https://ipdb.org/machine.cgi?id=3792 is the IPDB page for GoldenEye. There are two links under documentation, an English Manual and Manual with Schematics that may help. It’s a 1996 Sega Machine. Other machines made around this time: https://ipdb.org/search.pl?searchtype=advanced&mfgid=280