My opinion on this issue has always been that not all neutral sites are created equal, and thus we shouldn't see it as a dichotomy. There are roughly three types of neutral site games, IMO.
1) 50/50 crowd splits that just make WAY too much sense and thus create a special "event-like" atmosphere that is even better than a home-and-home for that specific series ... for us, this would be Braggin' Rights, and a football example is Texas/Oklahoma in Dallas.
2) Throwing a bone to a major market that has a large fan base there and/or is super important for the athletic program's success ... for us, this is the United Center game, and the vast majority of programs on the hoops side do this (e.g., Michigan/MSU in Detroit, Indiana/Purdue in Indianapolis, etc.).
3) Pointless neutral site games that would have been much better as a home-and-home series on the campuses ... for us, this is playing UNC in Nashville rather than Champaign, and a recent football example was USC and LSU meeting at Allegiant Stadium in Vegas than either one of their historic stadiums.
I'm firmly against #3, and I think that is where most of the hate for neutral site games comes from. I think #2 depends highly on context ... OSU doesn't get nearly as much mileage from playing a game in Cleveland as we do maintaining our United Center tradition. The neutral site games that fall in the #1 category need to be protected at all costs ... anyone who would ever want Braggin' Rights played on campus has simply never been, or he is insane! Those atmospheres are so rare and so cool.
TL;DR
For the Illini, I would want to keep Braggin' Rights in St. Louis forever and try to play at the United Center once per year, as both easily justify their existence ... all other non-conference games should be home-and-home, if possible.