I used to have the same opinion as you. That was until I started having a front left tire on mu Kubota B9200 that I could not keep air in. Replace valve cores, took it to the tire dealer twice, to no avail. 4 ply R1 tires are marginal for the load they carry w/ just the tractor, never mind the load in the FEL. I didn't like it, but eventually I slimed my tires. ALL my problems have since gone away. Gut feel is that the bead would not reliably seal, but who knows...I wouldn't use slime. Some tire shops will not change a tire when they find slime in them. It does work, but it leaves a mess when the tire needs changed.
John Deere does sell a tire sealant, I didn't care for it though. It was just very messy.
I have had valve stems go bad, so check to see if it is leaking.
As for driving on the flat, unless you see scarring on the tire, it is probably fine.