I am afraid that won't solve your problem. The spring on the core for a well Schrader valve is much weaker than the spring for a car tire core. You may have stopped any air from getting in.
It's possible that he put the right kind of valve in place, but I agree, that since he did not make the distinction, he probably did not.
Wiredhot, if you put a cap on one of those snifter valves for a tank with an AVC, it has to be one that passes air. The caps used for tires do not pass air, and in fact, the cap is supposed to be the primary seal for a tire. The symptom of the wrong valve
will be short cycling, although the wrong valve will work around your symptom of air in the water inside. The expected symptom of a failed seal on a snifter valve would be significant water leakage when the pump is running. Too much air in the house is not a symptom of a failed snifter valve.
If you have an AVC to release air, the snifter valve would admit air. There would be a check valve between the pressure tank and the special Schrader snifter valve. The snifter valve would often be on the upstream edge of the check valve casting. To read about this type of system,
"snifter valve" would be a good search term in a search engine. There is also a drain-back part of that system maybe 10 ft down the drop pipe.
A thing about your original symptom from your first post was to have an outside faucet teed in out in the yard. The problem with that is that the pump would not know when to start when you used the yard hydrant. The check valve would keep the pressure switch from seeing a pressure drop. But there could be some work-around, such as running water in the house while watering the garden with the yard hydrant. But given that hookup, the air from that yard hydrant would be expected. The appearance of significant air inside points to a failed AVC.
That "conventional tank" system has some advantage: it mixes air into the water, and that can help be rid of some H2S (sulfur smell). It also has no diaphragm or bladder to fail. It has some disadvantages too, and is less common today than the pre-charged pressure tank.