I did put the handlebar tape on my Schwinn.If I can get my act together we might have similar but contrasting single speed builds going on, mine will be fixed gear, hopefully path racer style.
my Nishiki Olympic frame has a similar very small head tube. I think it’s a 53cm. It’s just been a bare frame since I got it, but now I have wheels and a crankset ready to go, and some cool bars should be on the way.
if it’s too small I’ll strip some parts and build it up for my son as a single speed if I can fit a coaster brake rear wheel.