New Hoist... $100? ... Used Hoist, $50-$100?? Plus Shipping.
But I'm on a Super Slim budget... And I'm saving up to buy new tires... That's Why I'd like to cary my spare!!!

A New Chain, Shouldn't be more than $5.
Two Heavy duty Shelving brackets... I don't know $5??
2 small threaded shackles... maybe $5??
... ...A Cast Steel Eye Bolt and nuts, probably $5?
(OR I could use one I Used to use, for tying down my motorcycle, the one I no-longer own -- so... 800 lb Cast Eye Bolt - Free)...
One or two Clutch Springs from a Motorcycle that I no longer Own... Free...

--- Idea, Complete!! Budget Should definitely be less than $20.

I was able to free up my hoist, But One of the numerous Previous owners (which I Believe included a Salvage yard) cut the chain on the hoist, leaving only the chain links inside the hoist.. I was able to rotate the hoist and pull the chain out for comparison, the chain size is VERY close to One chain size Most Every Hardware store has, from one or another manufacturer. One store was willing to cut off a few links for me at no charge, so that I could test it out... whichever size chain it was, I believe a 3/16" ... the one that's much closest, has links that are too long, and it starts binding... So again, I (Was) sure it's a Metric chain, rather than an English one.

Thoughts? Ideas?? Opinions?? ... Offers???

It would require significantly different design, but I don't see why it shouldn't be possible, to lift a tire into place with leverage, and bolting it there, instead of lifting and securing with a chain and wheel.