[Goose]

You doubt that the program with by far the most national championships, the most final fours, the most WNBA #1 picks, etc. would be the most recognizable program in the country?

We can't know, because during the entire period where they did the vast majority of those good things they benefited from ESPN hype. Back in the "early days" one could argue Tennessee should have received the most hype but even back then it was clear that UConn was ESPN's preferred team. Many attributed it just to proximity. There is no doubt at all that their ESPN presence has helped recruiting at UConn, and that matters a lot. It also matters that the general public, who may not have been totally aware of many things UConn has accomplished are reminded of it continuously. This continues despite the fact UConn has not won a Championship since 2016.
There is no doubt UConn is a very successful program. However, it is a matter of degree. In the absence of ESPN I am arguing 1) UConn would probably have fewer championships/final fours etc. (recruiting advantages) and 2) the public would be much less aware of "how good the are". So,  IMHO without ESPN UConn would be one of the most recognizable programs in the country, but basically not so different from many other programs.

[Goose]

You doubt that the program with by far the most national championships, the most final fours, the most WNBA #1 picks, etc. would be the most recognizable program in the country?

We can't know, because during the entire period where they did the vast majority of those good things they benefited from ESPN hype. Back in the "early days" one could argue Tennessee should have received the most hype but even back then it was clear that UConn was ESPN's preferred team. Many attributed it just to proximity. There is no doubt at all that their ESPN presence has helped recruiting at UConn, and that matters a lot. It also matters that the general public, who may not have been totally aware of many things UConn has accomplished are reminded of it continuously. This continues despite the fact UConn has not won a Championship since 2016.
There is no doubt UConn is a very successful program. However, it is a matter of degree. In the absence of ESPN I am arguing 1) UConn would probably have fewer championships/final fours etc. (recruiting advantages) and 2) the public would be much less aware of "how good the are". So,  IMHO without ESPN UConn would be one of the most recognizable programs in the country, but basically not so different from many other programs.