It's hard to determine from your description how far the game is booting, if it is booting at all...

Some questions:

Other than the General Illumination lights, what else happens on power up?

Any coils kick?
Any Sounds?
Any Displays (I assume credit display) and what do they show?

Remove the backglass, open the head and watch the LED segmented display and the two LED's next to it...what are they doing during the boot process (be as descriptive as possible).

I'm going to guess the game is not completely booting, although if you can add credits, maybe it is...could be a number of things keeping it from booting completely, but the most likely is a bad IC socket somewhere, or a missing voltage from the power supply/board. This is where you should almost always start on troubleshooting a pin. Make sure ALL the voltages are present and within spec on the powerboard.

You might want to update your location as maybe someone local can help you out. With being new to pins and the game not booting, it will be difficult (not impossible tho) to diagnose over a forum. You should also get a copy of Clay's System 7 Guide for troubleshooting this pin as it has step by step procedures on what to check.

No batteries (or bad holder) isn't going to keep the game from playing. It'll flash the 2500 once, but as long as the AC power stays on, it shouldn't do it again. The fact that it does it again, means the MPU is resetting. So you're losing your 5V supply sporadically to the main processor. I would start looking at the 12V/5V circuit. You really need to crack open clay's system 7 guide and start troubleshooting the section on resets.

For solder that old (30+ years), I'd recommend you buy a cheap solder sucker (plunger type), remove the old solder and reflow some new. I did some of that on a Xenon recently (same era) and it corrected a host of problems.

Solenoids don't need replacing if they're working. But the sleeves and end-stops do need replacing. That's likely causing your weak stroke. The sleeves get gummed up, broken or melted inside the solenoid due to heat and that can cause weak stroke. The end-stops can mushroom over time due to repetitive contact with the plunger rod and that can cause weak stroke. I would replace those two items and you should be good to go. Before you order, remove the sleeve from the solenoid. Sometimes, they are so melted to the inside you end up destroying the solenoid trying to get them out, and then you will need to order a solenoid. A trick to get them out is get a socket (from a socket wrench set) that fits the solenoid hole and try to tap them out using that.