I just got a Color Classic which had the same problem--could hear a PSU noise when turning on the hard power, but no response to the keyboard power button. The caps actually don't look that bad (much better than my SE/30 in original condition) but I started looking at the vias around the crystal.

The via that dougg3 highlighted is fine on my board, but the one just to the left of it was corroded on the bottom of the board (pretty weird!). It's pretty obvious on that right side of C56.



Luckily that via just goes to ground, I added a wire to the battery ground terminal and now it starts up!