Find out what ATF he used. There are a couple different ways this could go. They do make "universal" fluids, but I'm not a fan of them - Toyota Type T-IV ATF has a specific viscosity and particular blend of friction modifiers to work with the transaxle. Unless the fluid specifically says - Toyota Type T-IV ATF compliant - don't use it. Some shops stock only one kind of ATF - to make it vehicle compatible - they add a friction modifier to the base ATF. This can have unexpected results - transaxle noise, slipping, overheating, etc. Even with a full flush, they can still have problems.
Top end flushes, where they siphon the old fluid out, is not the best way to do a flush on these cars. Better way is to connect to the transmission cooler lines, let the engine pump the fluid through, even better way is to use those flushing units that attached to the pump pickup. Those are more involved, have to remove the transmission pan and connect to the bottom - but it gives you a chance to clean any residue on the bottom of the pan and remove/replace the oil filter screen.
If the transaxle service is the cause of the vibration - that is hard to say. As there are a number of things that can cause the behavior that you are describing. But to put blame on the age and mileage, transaxle "expecting dirty oil" - that is completely unacceptable on his part. If the fluid is correct, should be no issue with the transaxle. With proper care and routine fluid maintenance and making sure the fluid never overheats, the transaxle should not have any issue. 200K miles over 8 years is nothing to this generation of Corolla's powertrain.
What is more likely to cause this would be the engine and transaxle mounts, serpentine belt tensioner, loose exhaust heatshield, or worn exhaust system donuts/gaskets. I wouldn't rule out the transaxle, but that is tough to diagnose. But since it sounds like you can reproduce it at a specific RPM (right before a shift, this generation has pretty even spacing between the gears)- I'd look into some resonance issue (something keyed to the RPM that is touching something on the chassis, transmitting the vibration.
