Start by checking these test points (game cold). These I posted in the Williams bulletproofing thread. Taxi was one of the test games used to create this reference table. Once you/we start to compare them, we can give you some idea of what you might need to investigate.

*** Williams System 11 Power measurements and test points ***

Logic power
MPU IJ17 Pins 4,5,6 = +5.03 VDC @8.1 mV AC ripple
U26/U27 ROM Pin 1 = +4.91 VDC

Audio power
MPU IJ17 Pin 9 = -14.04 VDC (Unregulated)
MPU IJ17 Pin 8 = +12.02 VDC (Unregulated)
Audio board U11 (U17 on Sys11B) Pin 8 -13.9 VDC @250 mV AC ripple
Audio board U11 (U17 on Sys11B) Pin 4 +11.8 VDC @50 mV AC ripple

Controlled lamp power
PS 3J4 pins 5,6,7,8 (Sys11A) or backbox bridge 6BR1 (Sys11B)= +15VDC to +18VDC nominal.

General illumination power
PSU 3J8 (Sys11A) or Interconnect board output 2P7 (Sys11B) = 4.5 VAC to 6.3 VAC nominal

Display power
3J5 pin 3 (Sys11A) or 3J2 pin 3 (Sys11B) = +90 to +110VDC nominal
3J5 pin 4 (Sys11A) or 3J2 pin 1 (Sys11B) = -90 to -110VDC nominal

Flipper power
Game specific@flipper supply board outputs= +65 to +75 VDC nominal

Solenoid power
PS 3J3 pins 6,7,8 (Sys11A) +32 to +35 VDC
Interconnect board F1/F3 (Sys11B) +41 VDC
Interconnect board F4 (Sys11B) +75VDC

NOTES
Reference test games used:
Williams Pinbot - System 11A/D-8345-549 PS
Williams F-14 Tomcat - System 11A/Rottendog WDP-011A PS
Williams Taxi - System 11B/PinScore PJ-12246 Rev. B PS

I disagree without more specific readings that his power supply is suspect. He read 4.98VDC on the 5V power supply rail when cold, which should be fine. It would have been helpful if we had confirmation of what the ripple voltage was. A higher than normal ripple reading would be a giveaway the 5VDC section of the power supply needs attention. Also, the measurement of the 5V rail at the CPU board was not mentioned. A significant difference between the 4.98VDC on the power supply board and the CPU board at the test point mentioned (actually, VCC at any of the logic chips) would indicate a problem - most often connectors but can be other problems. Another note, you would likely have random game reboots if the 5VDC was not good.

Not sure if a typo, but some of the reported values are indicated as positive values when they should be negative values. Again, I think it a typo, but it should be confirmed.

I would be curious to know what the flipper supply board voltage is reading (cold) since the weak flips reported at startup when cold.

See the schematics - insert power is rectified directly from the transformer. A problem with the 5VDC rail will have no association with this.

The transistors are biased to saturate or cutoff, in other words, acting as switches - either open or closed. The lamps would be full on or full off - there would be no dimming. Same goes for the transistors operating the solenoids - they would not be weak if the 5VDC had a problem. Those transistors simply wouldn't turn on. If the transistor switch comes on, the lamp should be at full brightness, if it is dim - it's because the lamp supply voltage/current is low. That comes from a secondary on the transformer that runs through a bridge rectifier....it's own pipe if you can think of it that way.

Have you tried another known good transformer? There is the possibility that even though you are measuring voltages that appear correct...there isn't enough current supplied under load until the game warms up. Also, make sure those output pins and connector out of the transformer secondary are perfect. That's what I would be looking into first if it was me.

Weak supply to the insert lamps would affect them all, but you could have additional problems specific to those inserts.

That's a function of the LED being a semiconductor. LED = Light Emitting DIODE. It won't light up until the voltage overcomes the diodes P-N junction.

7P2-7J2 connector out of the transformer. The only supply voltage not running through this output connector on the secondary of the transformer is the GI. If the GI is also dim, definitely would suspect the transformer (or something before it, input connector 7P5-7J5 perhaps), It might also be hard to discern if the GI was dim, unless it was really bad to be noticeable.

That's ugly. I'd have to question the long term reliability of some of those components, it looks like electromigration of the corrosives into some of the component bodies may be possible in this case.

You can neutralize the acid with vinegar, but reflowing everything is almost certainly a waste of time. Electromigration can get corrosion into the component bodies. Some of those components may be in need of replacement. It's hard to say which, usually every component that has been exposed for a period of time gets replaced. When this happens on the old Ballys, I use Ed's kit from GPE to replace the entire reset section group of components. The vinegar wash process is just to neutralize acid so the board traces do not continue to get eaten away, it doesn't actually "fix" anything.

I'm sorry to be the bearer of bad news, but if you are leaning towards replacing it - then that would be the way to go. If you like spending a lot of time desoldering and replacing components, make a list of all those components in the pic and any others that look hit with corrosion, do the cleaning, and go to town. An easy Bally is a good 4 hour evening (for me) to do a thorough corrosion repair. I've never done a Sys 11, and avoid corroded boards when I buy them. Just my honest opinion.