You really want to check the 5 volts for ripple. While a scope is by far the best method, you can do a poor man's version of testing with a decent meter. Put it on the lowest AC range and measure from 5 volts to ground--cheap meters will give you really wacky readings, like 10 volts for example. If you have another machine you can compare the readings between known good and unknown.

While not as cool as an eyelet press, you can put together the tools to do eyelets for about $50.

http://pinballrehab.com/1-articles/solid-state-repair/board-rework-test/230-repairing-plated-through-holes

That would do it.

I was taught the first three steps in troubleshooting are visual inspection, visual inspection and visual inspection. Over the years I would bet I've found 60% of board problems thanks to that advice.

You've got contact adhesion on a few of those pins--not surprising with a solder sucker. Heat them with a soldering iron and use a spudger (wooden dowel with beveled edge) to push them towards the center of the through-hole. Before you remove the socket give it a clockwise and counter-clockwise twist. That should break free any remaining solder.

You really don't want to damage any traces/pads on the ASIC. They're a bitch to fix.