If things were properly assembled, and assuming that there isn't a bigger underlying issue, the only way that the spindle could shift laterally (like when you knocked out the spur) would be if one of the two C-clip retaining rings wasn't fully seated. If the lathe is assembled correctly/completely, the bearings should be constrained by the machined pockets at each end of the headstock and those two C-clips. From your description it would seem likely that one of them is loose or missing.
BTW, I had some initial problems with my unit too and I got a little disappointed too. My perception going in was that everything would be perfect directly out of box. But they weren't. But Grizzly tech support was pretty good. So my initial problems seem to either be resolved or I can see the light at the end of the tunnel and am fairly sure that they will be resolved eventually.
Also, I joined the Green Monster group. I got great support and advice from its members. I consider myself fortunate to have found this great group and tremendous technical resource.