we should have a thread on headset play. there are YT videos that explain the mechanics of it. I think you have to tighten the top cap, because that's what compresses everything. basically, the top cap pulls the plug up (which is why you need space between the steerer & the top cap, that's the space that gets compressed).
When $hit's tight, your plug can slide up the steerer before there's enough system compression, so keep an eye for that if you have play.
I think people sometimes use plumber's tape (teflon) for loose bearings, electrical tape sounds ghettoer / maybe too flexy?
Btw, if your bearing really is swimming in its hole, then post a video, you may get a new frame. Some play might be acceptable, but if they botched it, they botched it.
It's safe to try the bike with a little play. Don't over do it, dont take it offroad, dont go down a mountain pass, be sensible.