This is the solution. After getting to 445-ish from WQs, invasions, weekly quest, etc (i.e. can easily be done solo), start with +2-4s, apply to every single group that has a tank and an open DPS spot, regardless of which dungeon. You will eventually get accepted. Nearly all sub-8 pug groups are successful if you only join groups that already have a tank. In these low runs, you’re practicing the routes, mechanics, and some affixes even if you’re not getting gear upgrades.
After successfully timing a few of these +2-4 runs, you’ll find you can use the same strategy (apply to every group with a tank) in the +5-7 range, and some will now accept. Rinse and repeat up to +8-9s, then once you’ve timed every dungeon in that range you can move up to +10s and get invites.
Continue with this strategy up to whatever level your personal skill can handle. When you get to a level where you’re dying to avoidable damage frequently or failing mechanics/causing wipes, back down a level or two and that’s where you should top out at.
Never join pug groups that don’t already have a tank, and avoid any group lead by Ragnaros players above +8, as these are the ones most likely to still fail even if you do your part.