Nothing about the shop's actions or recommendations seems suspect to me.
Chains need replacing the most often, cassettes less so, and cranksets much less so. Maybe 4 chains for every 2 cassettes and 1 set of chainrings, likely less chainring changes than that. This is assuming you're replacing the chain often and early when worn--you can blow through all three pretty easily if you just ride it into the ground. That said, with the mileage and time you've had the bike chainring replacements may likely be required and will almost certainly shift better. Given the relatively entry level nature of your bike, new chainrings (or often a new crankset--it's sometimes cheaper than new rings weirdly enough) of similar quality would be a pretty reasonable investment given the admirable amount of riding you do.
Your mechanic is likely recommending a sane course of action that DEFINITELY will work well. If you'd like, you can ask them to install a new chain and cassette and see if the chain skips on any of the old chainrings or shifts poorly, but given the mileage you have all new would be totally sensible without any testing.
Failing to find a tire isn't really a sign of trustworthiness or mechanical competence, it's really easy to miss things like that.
My viewpoint is informed but really biased as a professional mechanic. I have nearly never seen mechanics recommend needless service or try to rip customers off. A sign of a trustworthy mechanic is often that they will outline indicated maintenance and be clear at describing what will improve if service is completed, and what the consequences will be if its not and will outline a best-better-good lineup of options, wherein best is pretty much all recommendable maintenance, better is a good balance given the value and use of the bike, and good restores the bike to safe and rideable condition and/or avoids very expensive costs with inexpensive repairs.