Will these cheap $1-3 phone screen assemblies actually work?

When you see inexpensive stuff on Amazon or eBay, you are generally seeing bottom-of-the-barrel parts. These are the parts that pretty much every other major vendor has refused. Vendors do random checks of available lots and if the defect rate is higher than they'd like to see, they will pass on the lot. It doesn't mean they are all bad though and doesn't mean there aren't any good parts in those lots.

So yes, you could save money by buying this screen. If you have a tight budget or don't want to spend too much on an old phone, by all means order those parts. But there is a very real probability that you will receive a crap part and if you do, you may have a hard time getting an exchange or refund.