That is 100% symptomatic of bad capacitors on the logic board-a problem for which the Classic and Classic II are notorious.
Sometimes washing the board will bring it back to life for a little while. Otherwise, recapping really is the only permanent fix. The boards are big enough that it's not insurmountable, or there are a handful of different folks who will do it for you.
Honestly, though-I hate to say it but I had a Classic with bad caps and ended up just looking for one with a bad case that worked and swapping the boards as Classics aren't worth enough to pay someone to recap. I'm lucking in that my CII works on its original caps.
If I hadn't found a replacement LoBo for my Macintosh II, it absolutely would have been financially viable to recap and some others(IIfx, Quadra 840AV) are no brainers.
One last thing-if the battery has leaked, you can probably write off the board completely and just need to hunt for another.