Hello Will:
I would rent a carpet and vinyl removal machine, which has a scraper blade that oscilates and pushes the material ahead of it. You may have to go over it more than once but it is sure easier to hold the handlebar and let the motor do the work. Keep the blade sharp with a file or angle grinder.
I can't recommend a concrete covering; concrete needst to be 2" thick to prevent cracking, the epoxy floor patch products need to be applied to a clean floor.
Let us know how it turns out, you certainly have our sympathy.
Glenn
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