I would research schwinn catalogs and figure out when they stopped offering a 28" wheeled bike. That might possibly indicated the last time they were offered. Some of the manufacturers went right up to WW2. If I had the '30's catalogs I would try to give you a better answer...
Can you please show a close up picture of your rims. I can't blow-up the images you posted very large but it seems what you have are metal clincher rims, not metal-clad wood rims. I could be wrong but a close up image would help.
Those are some FLAT profile rims, I think @New Mexico Brant is correct, my metal-clad wood wheels have a much deeper profile.
I do have a similar pair of FLAT profile clinchers on a 20's Hawthorne Flyer Motorbike.