Not an exceptionally high end frame, but interesting nonetheless. How much does it weigh?

You might be able to replace the steer tube, but you might consider drilling the top and bottom of the crack. I like the idea of threading something down onto whatever part is hidden within the headtube.

The groove is necessary for the cable stop for the center pivot/cantilever brakes.

The crack is likely caused by overzealous tightening of the handlebar stem.

Unfortunately, without reinforcement, the handlebar stem will continue to expand the crack, causing the handlebars to loosen. Get enough nuts in the right place, and perhaps this could be prevented as long as they didn't interfere with the steer tube clearance.

Actually sleeving it will be tough. The head tube is about 1 1/4" ID. Your steer tube is 1" OD, giving you about 1/8" clearance on each side for the sleeve. And with the existing spread of the crack, it will be tight to thread anything onto it.

What I would try is to take a couple of generic top nuts from cheap headsets. Cut them flat on top with threads to the top.

FIRST Mount the lower cone, race, and bearing to the steer tube.

Now, carefully thread the modified nuts onto the steer tube so they end lower than the mount point for the upper race.

Grind the nuts round.

Assemble and hope for the best.