I dove deeper.
SW4 (the 'mode' switch), the only switch that works, doesn't connect the same way as the other five. SW4 is directly fed by 5v, and, when it makes a connection, is fed into the M430F4250 (the IC that I originally questioned whether it's good). It goes through two points, one is a via to the opposite side, through a 10ohm resistor in parallel with a capacitor to ground, and up to the IC.
The other long trace goes under the display, through a trace, connect to three resistors in parallel, and eventually to other stuff including C102 (capacitor near the ribbon cable). I'm assuming this is a more elaborate circuit as it needs to switch relays (as I hear them clicking when I push the 'mode' button.
The other five switches that don't work all go through vias to the diodes and resistors next to each other (R100, R101, etc...) and then into CH453D IC (16-Bit LED Nixie tube Driver and Keyboard Control Chip CH453).
First piece of curiosity is the buttons and how they make a connection. I measured the black dot on the backside of the buttons and they are resistive. I'm assumed they are 0ohms and connect the middle copper spoke wheel to the outer copper circle, however, that doesn't seem to be the case.
Are these capacitive connections or a known resistance that acts as a voltage divider so the IC knows which button has been pressed (my meter seemed to move around when I tried to measure the resistance so I don't know if it was moving due to them being capacitive or not getting a solid connection)?
As for the CH453D IC, it seems the five non-working buttons that go through the diodes and resistors and connect to this IC are connected to pins 1, 23, 24, 26, and 28. Looking at the datasheet, these pins have multiple purposes - from the datasheet:
Segment drive of LED Nixie tube, active at high level, Keyboard scan input, active at high level, built-in pull-down resistor
I guess the question is whether the chip is getting SCL and SDA, but trying to measure anything is way too difficult.
The only oddball issue I see is the buzzer appears to have been replaced due to flux in that area whereas no other portion of the board has flux. The buzzer gets 5v directly and the negative side is driven by a transistor that seems to be controlled by the M430F4250 IC. I'm sure this isn't an issue, but the thought had initially occurred that maybe the buzzer is trying to pull too much current from the IC (but it is driven by a transistor).