My grandad had a few greenfinch/goldfinch mules and some other hybrids, I don't think any of them were ever fertile though. I don't know much about keeping finches, but my grandad & my uncle kept them for donkeys years, mostly canairies, but also some captive bred native finches, such as bullfinches etc.
One day when I have more space & more time, I might give it a go, I used to find it fascinating sitting in the shed with my grandad, especially looking at all the beautifull red factor canairies, which always amazed me how they could drink couloured water & turn red!