Time for some recreational maintenance...
Fresh Schwinn head tube with little to no value.
Oops!
That was plummetting from shoulder height onto the door mat/concrete underneath, twice. Didn't feel that was enough for what I wanted to show was repairable. So, I hit it with the dead-blow hammer a couple more times. Nice flat spot!
Use a 1/2" drive extension and the largest possible socket that will fit into your closed up head tube. Upsize your socket as you go. Did this one with a 7/8" and 15/16" socket. Different manufacturers' sockets vary in diameter, and you will have to choose your own tight fit for the job.
Make sure the lower end of your socket wedges against the meatiest formed part of the head tube and you won't have to worry about other parts warping, just your flat spot.
Monitor your progress until "it looks pretty close" and don't go any further.
Wedge your original cup (or least favorite ugly one if you're really that worried about it) started on the dent side first, and then tap it in. Alternately, if you have an exerciser taken apart like this one, and have this lower cup still laying around, use it before the final fit of your original cup.
Even the part where the head tube met the concrete left minimal visible damage, a testament to Schwinn Quality!
As a side note, not sure why I hadn't used this socket setup for knocking cups out before. Plenty of weight behind it, perfectly aligned, and there's no "slipping through" before the cup comes out! Just hold onto the cup so you don't lose it and end up with a flying piece of steel aimed at your big toe... . .