Cart before the horse. And those 36 wins in 4 years were far from dominating. Bowl wins are bowl wins. But in the grand scheme of things they don't do much for perception unless you are winning one of the major bowls. 4 Liberty bowl wins, and now you have something. 4 Nola Bowls...that isn't going to move the meter...especially when it comes on the heels of no conference title or a shared conference title.
Why would we turn down an offer to better the conference we would play in? That's our peers. It's the only constant we have in scheduling. And those people we play CONSTANTLY are going to be the measuring stick for our program, not one off games that mean little in the long run. You are putting way too much emphasis on non-conference scheduling. If scheduling harder games outside of conference play was the way you became a bigger program, the entire G5 would be well on it's way to becoming P5. But they aren't. Because those games are close to pointless. You play the hand you are dealt.
If we get into a better conference, our main motivation should not be to schedule beatable P5's...it should be to dominate the conference we are in, and position ourselves as the most impressive team in said conference for conference realignment. That will outweigh any goofy, pointless, wins against supposedly better competition. People can read between the lines. By your estimation...La Tech's GARGANTUAN wins against middling P5 teams (at the time) should have them zooming up the conference ladder faster than you can say "Ole Miss as a team has quit on Houston Nutt." But...they ain't. They are not in any discussion. They are not on anyone's radar. They are slightly better off than we are.
Big dick posturing in scheduling sounds like a good idea in theory. But all it will accomplish is saddling us with a bunch of loses that won't really accomplish anything, and a few fluke wins that also won't really accomplish anything.