Was this everything the mechanic wrote? Maybe he did a compression/leak down test to diagnose the valve problem - IDK. But, if so he should have noted that, along with compression numbers. If not, it's nothing but a guess and could just as easily be spark plugs,wires, coil, or a bunch of other, relatively inexpensive things. But regardless of that, the thing that jumps off the page from this information is the P0700, which is the non-specific transmission code. It means that one or more specific codes were set, and are available via a scan tool capable of reading the TCM. This code means that there is a reasonable chance the transmission is toast, and will also need to be replaced. It's a major disservice to list a 1,000 repair estimate without mentioning that there may also be another 2K or so needed for a tranny replacement (in fairness, perhaps he did mention that to her).
It might be worth calling a couple transmission shops in your area and ask what they charge to read TCM codes, and give a diagnosis. There is a good chance the end is near for this Aveo, because it's almost definitely going to be a fairly big $$ tranny/solenoid repair. And unfortunately with these codes, no one will buy it, and she will only get salvage value for this poor thing . So the final answer will probably be just choose the drive-it-till-it-dies option, and then take whatever the boneyard is willing to give for it.