The building drain should never hold any liquid in it. If the piping is properly installed and clear of clogs the liquid from the toilet will flow all the way to the tank. A vent or lack thereof will not stop gravity, so your diagnosis is not correct.
A professional drain cleaner's snake with a properly sized cutting head will ream out the pipe to the original size...a typical homeowner's snake head is usually not that large and just pokes holes in obstructions. Because of the significant torque, a big head when it meets an obstruction, can generate some huge forces, and is best left to a pro to use. The snake can twist and break bones, lacerate fingers, tear things off, etc, when not used by someone with experience (and even then, if they have a lapse!). An AAV, should you add one where you already have an atmospheric vent, is a waste. Plus, it must be left accessible for replacement, since they do fail eventually.
I now come, hat in hand & bow humbly to the wise.
As before mentioned, I replaced the toilet & the section of pipe directly beneath & gave it a more aggressive slope & a vent. It worked great for 3 days. Then on the fourth day it wouldn’t flush at all, but the shower & sink still worked great…cut to chase, I went outside & dug up more of the pipe & found a huge root knot in the pipe where someone had knocked a hole in it years ago. I don’t see how anything was ever getting past it, but it was. It appears to me that someone had knocked a whole in the top of the iron pipe at sometime to snake it & didn’t close it back up. Got that out & everything is working great again. So, now I am going to replace all of the old iron pipe & add a clean out in the run out to the tank, which is about twenty feet from the house. I’ll do it all to code, except I’ll shallowly bury the outside clean out & put a galvanized steel plate on top of it, so it can be found with a metal detector. I’ll write that on the easily accessible basement pipe with a Sharpie. I can’t tell you how many cleanout plugs I’ve hit with a lawn mower.
I just wanted to follow up & say Thanks!!! To all of the people, that didn’t have to, but did try to help a po boy out…