Like a lot of things in life, there are choices which sound better than the current situation but a closer inspection makes things a little more complicated.
I will stick with the three sports I know a little about and leave the others to those in the know. In general there would be the probability of a lot less TV money. Also, a lot fewer games being available to televise unless the team USF is playing is affiliated with a conference with an excellent TV contract. You might get a local independent station to show a game or two but that is the extent of that. In men's basketball, I only know of 2 teams which are independent. They are Chicago St. and Hartford. Not only are those 2 not exactly name brands, I think Hartford has chosen to step down to Division III. I acknowledge the terrible fact that USF men's basketball postseason is an occurance similar to a comet sighting but I would argue if USF plays a similar schedule to the ones they did in those post season years as an independent, they would have only made it the year they were in the CBI. The baseball team of a couple of years ago would not have been able to even sniff the post season as an independent.
Football is a little different, of course, but there would still be issues we as fans would have to live with. Now, if this 12 team playoff format does happen, we would have to live with the fact that if the team somehow had a great season, they could win each and every game but still be left out. Obviously that is not the case if they did so in even a watered down AAC. Scheduling at times could be problematic. I seem to remember football Independents Liberty and New Mexico St. having to play twice in one season due to scheduling issues. Even as a new member of CUSA, USF had to play two 1AA teams that year. You can rattle off brand names you want USF to play but you have to get them to play you. It is not like the NFL where the league sets the entire schedule. You probably would see USF only being able to play 4 or 5 home games unless they decided to step down in class more than once a season. That is problematic if you wish to build your own stadium. Finally, I think most these days would be ok if USF did not make one of those lower tier bowls due to not being in a conference. That would be the reality though and we experienced that in 2002. That was a very good team and yet they were shut out.
The situation is not great at all but with the points above, I do not see how things would be better except for making our collective egos feel better. While I would not be against independence, I fail to see how it would improve USF's situation financially or as far as getting exposure.
I will add my thought or question on one thing Puc brought up which would have been interesting had it played out. If USF back in 2012 had stated they would go independent in football and stay in the Big East for everything else, would the Big East have accepted that? I have no idea at all.