I have a Gaggia Classic I've been able to maintain for many years but I'm having trouble figuring out what is currently wrong with it.

First, I'm impaired a bit because the steam light is just non-functional. I believe it blew out and it's very difficult to find a replacement set of buttons/lights in the US, and it has been fine just waiting about the time necessary.

However, many months after the light stopped turning on, a new issue appeared, which is that the steam only runs for a few seconds at first, and then after about 60 seconds, with no adjustments or changes, it starts again for about a minute, and then stops pushing steam for another 60 seconds. Eventually after these 60 on/60 off cycles, milk is up to temp.

Pulling a shot is no issue and running water through the steam wand has no noticeable issues. This only happens when steaming.

I've descaled, backflushed, replaced the solenoid, replaced the pump, replaced the steam thermostat, none of it has had any effect. I'm not sure how to troubleshoot further or what else it could be. Can anyone help me?

Wild first guess: One of your 2 heating elements is burnt out. While the machine will still heat to the same temp and hold that while idle, the single element alone can't keep up with the power needed for steaming.
Test: Unplug the machine. Pull all 4 connections off of the 2 elements. Test resistance across each element and test for continuity to boiler-body.

Secondly, What button/light set? New enough to have the three individual rocker switches? Or the older three switches in one "switch-box"?
WholeLatteLove will almost certainly have what you need.

I thought this was a great idea! I tested using a continuity tester and there was corrosion on the leads, so it was spotty. No problem, I replaced the boiler completely. The problem was improved (less scale occupying space in a clean boiler I guess?) but still hasn't gone away.

I have the system connected to something that measures power consumption and I notice hearing a click, seeing power spike up to 1-2kW and then go back down to baseline. I thought this could mean something - maybe faulty thermostat? So I replaced the brew thermostat (again). I still have the issue. I'm kind of running out of parts to change, all that's left is wiring and the switches?

As for the button/light set, it's the pre-2015 switchbox. I did ask WholeLatteLove and they said

I hate to be the bearer of bad news, but we do not. They have been discontinued from the manufacturer and the bad news is the buttons used on the Classic Pro are not retrofittable.

I also removed the thermofuse temporarily, just in case it was faulty, but no change from that either, so I'll put that back to avoid burning my house down, but it's one less thing to eliminate.

Posting for whomever runs into this later:

The problem here was actually the steam light itself. Absolutely the best explanation here on how the circuit works [https://comoricoffee.com/en/gaggia-clas ... iagram-en/] . The key is that the steam lamp itself influences the flow of electricity and it being on is not just an indicator, but part of the analog "logic" of the system.

As mentioned, the switches were not possible for me to get my hands on, even looking for pre-2018 Gaggia Classics for parts.

There is a video of a repair here [https://gaggiaclassicmods.blogspot.com/ ... -film.html], which if you're brave enough, you can try, but finding the properly sized neon lamp + matching resistor is as much of a challenge as finding a switchbank. Also, there was a risk (for me) of breaking it to the point where even the on/off steam wouldn't work and I wasn't willing to risk it.

The solution appeared this way: https://www.shadesofcoffee.co.uk/gaggia ... switchbank At original time of posting, it was not yet in existence, but I waited and just got it and it works perfectly and finally steaming works.