Most likely, the kind of damage you're describe came from driving on these tires with barely any air pressure in them. They will build up heat that way and fail in the area you described. I would stay away from run-flats, as they will ride poorly and not perform as well. I'm honestly not sure about the self-sealing tires. You can get a spare that will fit in the well under the hard cover "false floor."
I don't know how much of a variance there is in range among all the tires in the correct size. My hunch is maybe 10% at most, from the very worst to the very best. The important thing would be to keep inflation pressures on the side of the recommended range.
Believe it or not, for best performance in rain, you would really want a "summer" tire from Continental or Michelin. You want a very-high-performance tread compound that works well in lowish temps plus a tread pattern that evacuates water well. But those aren't really appropriate for best ride or long tread life. In your case, I'd look at the
"Ultra-High-Performance All-Season" category, and we can specify tires that have an energy-saving focus. Of the three that pop up at Tire Rack for our size, I'd avoid the Advanta. It's a no-name brand and it's cheap for a reason. The Bridgestone Potenza Sport AS and Michelin Pilot Sport All Season 4 are both excellent performing tires in all conditions. The Michelin comes stock on base-model Corvettes. Of the two, I'd go with the Bridgestone because it's on a very good sale right now: $616 for a set of four. These tires will sacrifice a little treadwear life and are generally more expensive than "Grand-Touring All Season" tires, but they will have more grip and perform better overall, and you implied that this is important.
If we ignore energy saving focus, the Kumho Ecsta PA51 becomes an option. It's a great performer in dry and wet, only sacrificing snow traction (which you don't need). They are closing it out, so you can get a set of four for $428. You'd have to save an awful lot of energy to make up the price difference between these tires and the Bridgestones above! If it were me, I'd choose these.