I work as a bike tech myself and if your telling your story exactly how it happened than to me it seems they took you for a bit of a ride.
Tire wear is really subjective and even a "newer" tire can wear considerably for a variety of reasons or begin to bulge/sluff from incorrect psi pressure, sitting in the sun too long, etc etc. So I can't say much on that because It CAN happen...

but the chain issue seems like complete bs to me. as a tech, I can perform a basic tune regardless of a slopping chain. you can still properly align the rear and front derailleurs just as well, albeit with the expectation that a worn chain is not going to be as precise and efficient as a new one. Chains can wear quite faster than one would expect, though, considering the quality of the materials its made out of, the abuse the rider puts it through (how long is he or she riding big/big, small/small....) and so on. At the very least your tech should have shown you how he measured the wear and definitely should NOT have forced you to buy a new chain from them and pay for install to finish a tune... that's just ridiculous.