The recommended rim size for tires is typically wider today. We usually have to go by the minimum rim size quoted by the tire manufacturer. The 19" rear is normally a WM2, and the 18" rear is normally a WM3. A WM3 is 2.15 inches. For Avon Roadrider AM26 rear tires, the only size they offer with a minimum rim size of 2.15" is the 4.00-18. This is the one I recently mounted on my Egli. The overall width of the tire is 4.6", and it has a load rating of 617 lbs. For the Avon Roadrider AM26 universal model, there are 4 sizes available with a minimum rim size of 2.15": 90/90, 100/90, 110/80, 110/90. The tire width ranges from 3.8 - 4.6". The 90/90 actually has 2.15" rim as the recommended size. Those universal tires, as mentioned by Glenliman, have less tread depth, and the load ratings are all lower, between 430 - 567 lbs.
For WM2 19" rims, the rim width is 1.85". For the Roadrider, there are no rear tires listed with a minimum rim width of 1.85". For the Roadrider universal, the only choice is the 3.25-19 with a load rating of 467 lbs. Looking at the Classic section, there are choices in the Safety Mileage, a 3.50-19 with load rating of 507 lbs, and the AM9 (old Roadrunner) 4.10-19 with a load rating of 567 lbs. The AM9 does have the 7" tread depth, although it is listed as a universal tire. The SM 3.50-19 has a tread depth of 8.1".
If you are willing to forego the minimum rim width recommendation, there are a few other Avon choices.