A flat increase to the chance to overmeld probably isn't going to happen, for the simple reason that it would be a really simple change to make so the reason why it's the way it is is almost certainly a deliberate game-balance decision.
I support the concept behind your second idea, but the execution needs work. If you increase the chance every time then you're affecting the average number of attempts it will take. Using a back of the napkin probability estimate, if you start with a meld that has a 10% chance of success and increase it by 10% with each failure, you'd average about 4 tries per meld (since 10% + 20% + 30% + 40% = 100%. Not rigorous calculation, I know, but ballpark is good enough for the sake of discussion.)
What I'd love to see is the ability to put multiple materia into a single meld attempt and increase the odds that way. So for a 9% chance you could opt to jam 11 materia at the slot in one go to get a 99% chance of success. This would consume all of the materia regardless of what the RNG roll is, so you'd still be using up the same number of materia as you otherwise would on average, but it lets people opt out of the risk/reward aspect of trying them one at a time. (You can't get the first-try awesomeness that happens periodically but you also avoid the 37-failure streak.) If they wanted to encourage people to use the current method and take risks, they could make each extra materia have diminishing returns when adding to the success rate (to some minimum point, say 5%.)