would you replace those because of the burn marks underneath?
Use the DVM in diode mode (beep) to see if they work, if they do, no need to replace.
http://www.youtube.com/watch?v=fbSi_X7Ew3YHere a vid of how a diode works...
http://www.youtube.com/watch?v=eJdGYtU7OZc&feature=relatedback to this one! i went to my mother's place to examine the board a little bit closer (and look for that test switch which is supposed to be there somewhere). let's examine the only switch on the board in detail: as i've already said you can read 1/p and 2/p next to the switch, but GT10 is a single player game. what could the function (if any) of that switch be? suggestions?
RTFM

up next, i examined the time potentiometer closer and realized that it is broken - literally!

however, all of the contacts seem to be good, so if i'm not mistaken it should work, but nothing changes at all when i turn being turned!

suggestions?
Yes you are mistaken

the pot is using the contact on the wheel that is completely cut off.
the 3 connections dont have any contact with eachother so the connection is "open"
Replace the pot and try to crash again (withing the time that also will work again i think after replacing the pot)
coming up tomorrow! can you tell from the manual how this was supposed to work / be wired?

when i see those solderpads, i think there should be the same switch on that board as the P1 - P2 switch.
This is a switch that connects pin 1 to pin 2 and pin 4 to pin 5 in the "test" position (see my blue lines) and for normal position it connects pin 2 to 3 and pin 5 to 6 (see my red lines)
be sure to remove that "fake switch" if you want to try the test switch, otherwise it will not work, and i dont know what the consequences are when they are all connected by that.. (i mean pin 1, 2 and 3 will be all connected, and pin 4, 5 and 6 also)
I would start with the pot, my guess is that that is the source of your problem.
Goodluck !