From a technical point of view this problem can be many things until one by one they are ruled out.
When a tape doe snot play in my decks and I do fix a lot of decks, I test the tape by turning with finger or a old BIC pen outer body that was shaped perfectly for this purpose. If the reel turn easy then it is not a slip sheet problem or a tape pack problem but might actually be stiction of the tape.
Back in the 90's an Indian person came to many tape manufactures and offered a new binder that was cheaper- dollar signs got in front of their eye and like in most cases where bean counters are listen to the final product was a disaster for companies like Ampex, Scotch and also Sony.
Those companies that smartly told the guy to take a hike were BASF and Maxell. What the results were from such bad choices were things we now know as sticky tape in the reel to reel business. Cassette tapes from Scotch were also known to squeal and not be able to be played. I know because I had some and although TDK and Maxell tapes played on my Teac A450 perfectly the Scotch would not.
You may be dealing with a stiction condition and a sign of that is the screeching sound as the tape no longer slides across the head but get stuck to it. You might try a Nakamichi deck that has pad lifters but there is no guarantee that it will play there either. It is just that the head contact is made worse when pads are involved.
I also had a deck returned to me as one that would not play commercial tapes. It was found that the cheaply made commercial tapes had sticky or decaying pads that caused high wow and flutter and causing the cartridge to be elevated in the holder lessened the effect. Onec can NOT use commercial tapes for evaluation of a decks worthiness as most of the time they are garbage. I put a Maxell UR tape in the deck and it played
perfectly fine without issue so if you can use the least made media and get better results than the commercial tape- well you can determine where to place the commercial tapes quality wise.
fast wind only make a fix for tapes that are poorly packed or with poor slip sheets. In these you can transfer a tape to a new shell to recover the audio but it will do nothing for stiction.
Some good results have been had with Nu Finish car polish on tapes I know were sticky. The problem is that cassette tape is so thin and delicate that you would need to build a special mechanism to process the tape so that it was not damaged. The process is to remove moisture with an alcohol and then treat the oxide surface with the surfactant that is in new Finish that seals the tape from moisture. it is unknown how many times a tape can be used after treatment but I have had reel tape alignment tapes that have passed at least the 50 play limit before needing another treatment. The surfactant does wear off apparently. It is easy to treat it again.
Thanks, I learned so much from your post. The story behind why cassette quality decreased in the 90's was interesting. Also wish nakamichi players did not cost so much.