If you're only getting 3.5V at TP1, then your game is neither coming up nor completely dead. That's probably *the worst* position to be in, because then everything is in an undefined state, neither off nor on.
Leave the under playfield fuse blown (or pull it) until you get your +5V sorted out. You'll just be repeatedly shooting yourself in the foot until you find out the cause of your low +5.
A melted connector cannot look too bad yet still be melted. It MELTED. It's already had WAY too much current and heat put through it. Stop being lazy and replace it AND the pins it plugs into. Since you're going to be pulling out the power supply anyway, fix the pins that were soldered directly to the power supply. That's a hack that works, but as you now see winds up robbing peter to pay paul (no serviceability).
For now, you need to work on just the power supply and get that squared away before you go any further. I'd pull ALL the connectors on the power supply board until you find the source of +5V not being +5. One thing you didn't mention checking / changing was the bridge that's the source of +5V, nor the 11kuF cap on the +12V unregulated. If either of those is blown it'll wind up trashing the other, and causing problems holding up the +5V.