Failed and worn valve bodies; are the typical problem, valve body replacement with a rebuilt unit, has fixed many members cars. Check out this vendor; for explanation of typical 01M problems, a diy page installing a valve body and valve body rebuilding services. There are many vendors; that offer rebuilt valve bodies but his pages, have allot of excellent info about the 01M. You might contact him and discuss your situation and see what he recommends.
Again I do not understand transmission shops and shops in general.
The problem is the general word on the street is that ALL VW automatic transmissions are bad so nobody wants to touch them.
Transmission shops prey on everyone. At the end of the day trans shops will ONLY perform entire rebuilds, they RARELY "repair" a specific problem. The normal MO is we will inspect the transmission for say $75-$150, what they then do is drain the fluid and pull the trans pan. They they tell you the trans pan is full of metal and you need a complete overhaul. If you agree to the overhaul you will need to pay the inspection fee.
The problem is EVERY automatic transmission has metal in the pan. The question is how much metal is acceptable?
If your transmission has 161k miles, then the transmission was not bad to begin with. Will the transmission last another 161k miles without a complete overhaul, probably not. But at the end of the day, the valve bodies and solenoids do have problems on these transmissions. The valve bodies can be overhauled, usually there are a few bores that need to be reamed oversize and new spool valves installed. Solenoids need to be tested and replaced as needed. This will usually extend the life of the transmission for some period of time.
A smart/informed person can judge from the amount and type material in the transmission pan how badly worn the internal components are and how likely the valve body repair will be to extend the life of the transmission.
Having the valve body overhauled is around $450-$500 typically, then you need to add about 3 hours of labor and fluid and filter into the repair. A typical valve body overall repair is usually in range of $800-$1100 depending on the location in the country and local labor rate.
If it something that people can DIY as well. It is not a hard job, just tricky to disconnect the solenoids and keep track of the wiring.
Like anything it is cost/benefit trade off. Is your car worth chancing another $1000 or so to buy some more time?