@Mike Palmer Plumber from Wisconsin here. It sounds to me like it is a tree root issue. From what I read it sounds like you have cast iron pipe underground which tends to rust out after a while. Especially with the house being built in 59. Trees seek abundant water so even after the roots have been snaked and jetted, a short time later they grow back. For this, like Mindy also suggested, either replace the lateral going from the house to the street, or have a yearly maintenance plan. At some point though, it will need replacing.
Obviously before going to any extreme, the best thing to do is, like others suggested, spend a few hundred and get the drain snaked again. (To reduce water/backup.) Then send a color camera down. If its a decent camera they should have a locator with it and can tell you almost exactly where the blockage is and you'll be able to see exactly what it is.
Also, assuming its not a broken toilet flange, which is what you bolt the toilet to, clearly your wax ring isn't sealing correctly. The plumber will replace that for you after the inspection.
Good luck my friend and keep us all updated. Unfortunetly, with cast all over the country coming to age, many other investors will run into this problem.