How to check if the gear shifter or the gear itself is broken?
This does not answer your question, but it's too long for comments.

Summary: take it to another bike shop.

Details

This is an internally geared hub (IGH). I cannot tell what make it is; the ones I know of are Sturmey Archer, Shimano Nexus and Alfine, and SRAM.

The whole bike has been left to molder - the chain has rust, and the tires are flat. The chain should be fine with some oil.

The question is - why was it left to molder, and why was it sold? My guess is because of the gear problem.

Advice

See if you can identify the brand or make. Someone might be able to help more if you can tell them that. Once you know that, you can also look for a bike shop that knows about those particular hubs.

IGH's can stop working for fairly trivial reasons, and often the new parts are not very expensive. The main cost is the mechanic's time.

While you could DIY, the fact that you asked the question suggests that that's not a realistic option for you. There is no guarantee that after taking your first IGH apart you'll be able to put it together again. There are lots of springs and pawls and bearings and gears in there ...