The extremely tight fit upon inflating the tire is a very long-time SCHWINN thing, and I always dreaded having to ride home on a thumping tire because my pocket pump would not quickly enough inflate the tire to make it seat after fixing a puncture.

When at home, I always use my rubbing alcohol spray bottle on both beads right before inflation. And I still have to deflate and re-flate a couple more times before getting full seating half of the time.

The upside to all this is that when the rim is used on a tandem, the blow-off pressure of a normal tire will be waaay up there.

Things that can help include:

Make certain that the rim tape is in and stays in the most central spoke channel. Must not cover the inside ledge!

Use 70% alcohol or suds on the beads before inflation, and inflate quickly to max rated pressure or slightly more if the bead is mostly up there.

Don't let the weight of the bike or even the wheel(!) rest on the deflated tire before inflating. Allow the tire to center itself with only it's own weight acting on it, or keep the wheel/tire horizontal.

Tension spokes to a good 80kg all the way around, possibly more if the nipples are well lubricated. Many Schwinn wheels appear quite under-tensioned making the wheel diameter bigger.

Avoid Continental tires on Schwinn rims.

A bead that is sitting low at one very small length of the bead can often be twisted upward using both hands on the tire. I even got this to work in the field once.

I suspect you could use folding tires on these rims w/o fear of blow-off, but I can't recommend this. I did run Michelin Hilite-Tour 27" folding tires on a set of Araya rims and it worked up to about 70psi, this before I knew better as it were. I did actually ride about 800 miles on that setup during winter training months in the late 90's.
I have a bundle of Folding 27" Pasela tires stashed here that I will try putting on my Varsity one day soon. I suspect that they will safely inflate and keep my 145lbs off of the pavement, after I first test them to 85+psi in the name of science.

That's everything I know about this topic.