The rim should sit on the rubber ring on the tire changer with the alum cone on top to center the tire. That will help prevent rotation when using the HF iron tool. I lube the beads and rim with dish soap solution when removing and reinstalling. Having the solution on hand also helps with checking for leaks at the end. My luck with patches is like yours. PaulL
FWIW. I had a bad snowblower tire; flat when delivered new. Inflating it would usually hold through winter. Then when doing prep for an incoming storm, it wasn't holding air well at all. It was leaking all through the sidewalls. With nothing to lose, and hoping to get the tire to hold long enough to get past the storm, I put in some Slime, laid it down flat overnight. The next day, I reinflated it and laid t down flat on the opposite side. That was 2 seasons ago. It holds air better now that when new.
One little burr on the lip of the rim will do it. I sand the heck out of the bead and lip before putting a tube in. Also as you said you need to find the source of the leak and remove it. A small remnant of a nail or screw will ruin your day!
I hate tubes ralph. I end up just putting atf in the tires because it swells the rubber in dry rot cracks. If it’s a bead leak, it gets slobbered with Bead sealer. If it leaks after that, it gets a new tire.