Why does it do this?

Try the disc drive solution to see if anything can be done for you.

support.xbox.com/.../disc-drive-solution

Sound to me like your laser start having problem

Once again, cannot edit my post. I'm not sure it's drive failure, becuase it did recognise a) there's a disc in there b) it's a game or c) it's CoD4

But it just will not play

Both have scratches

CoD4 has a tonne of them, none of them are deep (only light scratches, but there lots of them). Not one inch of it hasn't got a scratch, so that more than likely explains that one. Given its age, not surprised

Tekken Tag has some, a surprising number for how long I've owned it. Got older games that are more than a year old and they look in better shape

Probably not.

Have you checked the discs for scratches? If they are scratched perhaps that's what's causing the issue. If you have friends or a video store nearby that you can borrow the games from, do that and install them on your console from the borrowed one.