I've also done the weld patch!! When we swapped our Suburban (box somewhat loose) for the Excursion (loose then!!), welded, and towed for about 10K miles with Jayco we had then. Last spring, I redid the welds and grinding for a tighter fit with our new Timber Ridge. After aboot 3500 miles, back to slop. Just as yours, it threw off my DualCam adjustments by at least 1 link.
It looked like the welds were putting more force over a smaller area on the box. The top of the receiver at the rear was bowed up maybe 1/8" +. Couldn't see the front bottom, since it is boxed in on the Ex receiver. Along with that and 12yrs of rust and repainting, I replaced the receiver with a Curt receiver. New one seems to have more "meat" to it, so hoping that it will stay tighter longer. Reground the shank back to normal shape, fits nice and tight now. Haven't used yet, first tow will be in early Feb down to the factory for some repairs.
I've got the feeling that Ford doesn't use the best metal in the receivers, and over time they tend to distort. Rebuilding yours with a new 2" tube maybe a good way, but also check into a new one from another mfg. I wouldn't go to a 2-1/2" unless you really plan on a larger trailer or truck. I've noticed that the larger shanks are pretty spendy!! AND, as you said, much more weight!!
See you on the ORV site!!!