Ahhh, the all infamous Disc Read Error. There are many possibilities to the root of this problem in, the most common is displayed
here as well as the solution to it. I have gone through that problem multiple times in fact and fixed it in that fashion.
However after a while what started out as discs not being able to auto start escalated to the point where CDs, DVDs and some PS2 DVDs could not be read at all. I found out that this could be caused by a voltage problem where the disc drive doesn't spin the CD and/or DVD fast enough. The way to potentially remedy this problem can be found
here, however like the faq states trying to fix this problem doesn't come without risks that can probably permanently damage the PS2.
As for what I did with my 1st gen PS2, well like you I ended up buying a V2 since when stopped reading virtually anything all together was the day I bought Gran Turismo 4 as well as just the usual being fed up with the constant maintenance. However, I didn't throw away the other PS2 since I made a deal to fix it with one of my friends and give it to him. Perhaps you can do somewhat of the same, but instead you would probably rather sell it or something.
Also, LandShark there is no GTA out right now for the PSP. Rockstar said it probably won't be out until August and now with that whole "Hot Coffee" tweak exposed, it might be delayed even more.