Maybe you can borrow a friends machine first to check it, as my TV (also from 2001 - A Sony model of some sort) is labeled as a 50Hz TV on the back but does offer PAL60 support. It's worked fine with my 360 and my Dreamcast before that. I could be that your TV will work despite what you've read so it might be worth checking if you can borrow someone's 360 or Dreamcast even.
Of the 9 games I have for my 360 only 2 are 60Hz only: Gears Of War and Pro Evo 6. Both state so on the back of the box and also have a big-ish 60Hz symbol on the fronts too. I guess this is due to time it would take to optimise the game for PAL 50Hz signals and means they can release the games quicker without the hassle and expense. Both of these games in my opinion were probably rushed out, Gears for a Chrimbo release and Pro Evo being pushed out by Microsoft (with considerable less content that the PS2 iteration)
As an alternative, you could always get a VGA cable for the 360 and hook it up to a PC monitor if you have one. Some games might not work though, Xbox Halo 2 being the only example I can think of.
EDIT: Halo 2 does work with VGA on 360. I bought a cable a couple of weeks ago and the game boots up fine (not played it though). From what I've read Halo 1 might not work with the VGA cable but I have no first hand experience as I don't have the game.
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