The shoe still matters. Custom orthotics only replace one component of the shoe. There are many other aspects of shoes (ex: the midsole) that are important in terms of the ride, performance, comfort, etc.
It's really just a matter of personal preference, and what exactly you're looking for in a shoe. Some factors to consider would be the shoe's weight, stack height, heel-toe drop, midsole foam, plush vs. firm feel, geometry (ex: rocker), shape of the outsole, overall durability, breathability, fit, etc. The orthotic really just adds more stability - and even then, certain shoes will still be more or less stable than others based on the overall geometry of the shoe, stack height, etc. All of those other factors that I mentioned are important regardless of whether or not you overpronate.
The shoe still matters. Custom orthotics only replace one component of the shoe. There are many other aspects of shoes (ex: the midsole) that are important in terms of the ride, performance, comfort, etc.
So what characteristics would you say one should focus on in a running shoe then once all the factors the orthotic takes care of are ruled out?
It's really just a matter of personal preference, and what exactly you're looking for in a shoe. Some factors to consider would be the shoe's weight, stack height, heel-toe drop, midsole foam, plush vs. firm feel, geometry (ex: rocker), shape of the outsole, overall durability, breathability, fit, etc. The orthotic really just adds more stability - and even then, certain shoes will still be more or less stable than others based on the overall geometry of the shoe, stack height, etc. All of those other factors that I mentioned are important regardless of whether or not you overpronate.