The first lathe I built I used an old grinder arbor as a spindle too. Until I discovered what it was and started using it as a grinder. Then I had to make a new head and just dedicate that grinder arbor as a grinder again. I still need to get some decent bearings for my homemade head someday. I actually completely rebuilt it at that point. i.imgur.com/7KTgP1C.jpg
I did not have all this stuff lying around either. But I got everything for free from friends and coworkers. Its not that hard to find a few bars of iron and a motor.