Even simpler than that, do you have to hold down record and press 'play'? I don't remember seeing many decks that will go immediately into record mode, as a safety feature.
A piano-saurus transport will not allow the record key to be pressed down and locked if the record safety tab is out on the cassette shell. So it's something else.
I didn't see the OP confirming if he tried what jpmorris asked you about in post # 3, so double check that too.
And yeah, used older consumer cassette decks are a dime a dozen on Craig's list. So just find another bargain unit somewhere. It's not worth repairing.
A piano-saurus transport will not allow the record key to be pressed down and locked if the record safety tab is out on the cassette shell. So it's something else.