It's just a galvanized tank. It contains no bladder and it has no
schraider valve on top.

Then the plumbing/well guy is probably correct about the tank
water-logging. However I would be very nervous about his suggestion
to drill a small hole in the pipe a few feet below the well head. I
would first try to install a schrader valve at the top of the
galvanized tank. Of course that would require you to periodically
check and recharge the tank air. If that is required too often then
you probably need a new tank.
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any tank with air and no bladder will eventually get "water logged" because the air will dissolve into the water over time.

I don't know how long this normally takes but I thought those type of tanks do need to be refilled with air periodically and therefore should have a valve of some kind.

If your tank water logs faster than normal, it may have a tiny leak that allows the air to escape but too small to let much water through.

m