I'd heard some rumor about a nintendo disc which might be useable for testing the Wii, but I honestly can't find any definitive information about it.
I would do this. First, I would get an original game (rent or borrow one if you don't have one), and see if it will read this properly. Play it for a while and just see if everything works as it should. If it reads this fine, the problem is probably related to media that the games were burned to. If this is the case, then burn a game (as slowly as possible) to a different brand of disc and then test it out.
If the original game still doesn't work properly, it's probably likely that there's something wrong with the drive. Since the modchip doesn't consist of moving parts, it's very unlikely (though possible, I'm sure) that the modchip would only work sometimes, usually they either work completely or not at all. The drive is made of moving parts and they are far more likely to work occasionally or work for a certain period of time and then fail. These can be replaced with a guide and some patience.
If the Wii is actually shutting down, it's probably the power supply.
I'd probably look to get my money back if it was anything but the disc media.