I'll have to take a photo but I'm about 99% positive my blacksmitthing club has a bader with variable speed running off single phase. I think it's come up on here before. Certainly not the norm but I do think it can be done. I'll take a photo next time I'm out there.
There are other methods. It also depends on what you are trying to make variable speed. A small fan for example is different from a 3hp grinder. One of the more popular methods I'm aware of are special v-pulleys. Basically they expand and contract changing where the belt rides in the pulley wheel. My shop smith and powermatic dp had variable speed done this way. To your point though, how we do it with a 3ph motor and VFD is about the most effect and cost effective way that I have seen.
As far as the OP's issue, it's unfortunate but as someone else pointed out you bought cheap and got cheap. I know you realize this now, but for all the new guys on here maybe some will avoid a costly lesson because you shared this. Hopefully you can send it back and get your money returned.
-Clint