The Idle valve is built into the throttle body. And don't push against the motor on the flap. It's an expensive mistake

Any faults recorded? Also did they erase learned values before running basic settings. Did it say "ADP OK" afterwards? Is the throttle valve clean inside?

Inlet gaskets can go. But rare. Brake servo pipes used to crack, there was a recall so I'm sure it will have been done by now.

Does sound like the throttle body though.

Quick thing is the throttle cable too tight? That can mess it about but again its rare.

Next thing would be a compression check. Oh and make sure the ignition timings OK. Probably is, but can't hurt.

I've had a Polo doing this to me a long time ago, can't remember what we did in the end. Probably just disconnected the battery for 20 mins