During BYU's independence era, they played six teams four times or more. From those six, I've widdled it down to four teams that BYU is glad they don't need to play on a regular basis anymore.
Utah State
BYU is 8-3 vs. Utah State during independence, so you might wonder why BYU would be happy not playing a team that they have a very high chance of beating year in and year out. For me, it's the risk vs. reward factor. Utah State isn't typically highly ranked or considered a strong team so if you go in and beat them, or even crush them, it was the expected result and doesn't get you much, or any, increase in the national standing. Now, if they happen to pull off the upset and beat BYU then it hurts...bad. So the risk vs. reward doesn't match up.
Along those lines, Utah State has been the team that gave Taysom Hill two of his season-ending injuries while he was playing for BYU along with one season-ending injury to Jaren Hall as well. This "rivalry" also has seen the dark side of rivalries when in 2021, Utah State thought it would be a good idea to move BYU to the home sideline so they would be right in front of their student section. Well, when you get that in combination with a 34-20 rout of the home team, you get the student section throwing garbage and bottles at the BYU players. Given the small stadium, some students were only a few yards away from the BYU players so it's hard to say that the result was surprising.
Idaho/Idaho State
I'm lumping these two together. The struggle with independence was getting good, quality matchups late in the season. Most Power 5 programs are in the thick of conference play in November, which left BYU picking from the bottom of the barrel for teams to play at that point in the schedule. This usually left BYU with Idaho or Idaho St.
These were meaningless, filler games that were played when it was bitter cold which meant there usually weren't many fans fighting to get tickets to these games. The only silver lining was that these were usually blowout wins by BYU and gave the younger guys on the team a chance to play and get experience.
Umass
Along the same lines as the Idaho schools, Umass became a regular November game for BYU during independence. Umass also went independent like BYU in 2016 which meant they were an easy pick to fill game slots in late-season matchups. These matchups were typically a blowout, except for BYU's 2017 season, which was one of the worst seasons in BYU football history, where Umass beat BYU 16-10.
I would be happy if BYU never played any of these teams again. I am very happy that BYU is now in the Big 12 and can have some exciting, meaningful matchups in November again! Tom Holmoe, if you're reading this, please never schedule Utah State again.
Independence is what kept the Y relevant. Hand picking opponents along with some good upsets kept them relevant during the independence years. After watching last nights game unfortunately I think my cougars will now be forced into Big 12 irrelevancy as a bottom feeder of a power 5.