This issue should not be difficult to figure out for any real mechanic. I highly doubt your transmission is damaged internally as that condition is pretty rare… Possible sure, but they can take a lot of punishment before failure.
If while the engine is running it is difficult to put into gear… Yet you can get it into all gears with the engine off.
The VERY first thing to look at or suspect is the clutch system… Which consists of the clutch master, the clutch and pressure plate and the clutch slave. Basically all the components that make up the clutch.
Commonly the clutch slave fails…which doesn’t allow you to disengage the clutch and get into gear with engine running… Problem is this symptom can also be caused by the other components of the clutch. Up to the mechanic to figure this out…
In VW and Audi there is a VERY very common problem of the DMF failure. The Dreaded Dual Mass Flywheel… Its made of 2 pieces and those pieces begin to come apart if not completely fail all together… They are failing all over the place on VW and Audi … including my own VW GTi.
Any mechanic worth his salt can figure this one out… what confuses most lesser ones is this DMF failure. It will produce the same symptoms as you have AND it will make noises that worry or confuse some “mechanics” I am betting he is not recognizing the DMF failure noises and blaming them on internal trans problems which take a lot to damage internal trans…and I mean serious abuse so that should not be suspected till much much later.
I bet your issue lies somewhere in this realm. In order of suspicion, I would go after the simple less expensive items first (naturally) then move on to the DMF failure (COMMON as hell). With this in mind it could guide any mechanic to the solution. Just know DMF failures are COMMON… Look it up. If it is your Dual Mass Flywheel… You will be replacing everything anyway… New Clutch, Slave, Pressure plate and DMF…
Did I mention the DMF failure was common ?