Until two years ago when the College Football Playoff was implemented, there were plenty of ways teams could be awarded a national title because there were so many different polls. Different programs have different philosophies when claiming National Championships. Let's outline some of the instances of this and offer an opinion at the end.
This author went to college at the University of Missouri. While Mizzou is not currently one of the top programs in college football, there certainly is a level of tradition at the school. They have been awarded two national titles in football—one in 1960 and the other in 2007. But they claim neither.
In 1960 the team went 11-0, according to Big Eight records. They actually lost to Kansas, but later the win was vacated by the Big Eight because the Jayhawks used Bert Coan in the game, who was an ineligible player. The Tigers defeated #4 Navy in the Orange Bowl that year, and were the highest ranked team in the country that won every game; finishing #5 in the nation. Second ranked Ole Miss was also undefeated but they had a tie with LSU. However, Minnesota is officially the National Champion from that season, because they were ranked #1 in the final AP poll. The Gophers lost to Purdue in the regular season and to Washington in the Rose Bowl.
In 2007 the Tigers went 12-2, with both losses coming to the Oklahoma Sooners, once in the regular season and the second in the Big 12 Title Game. That was the year LSU was awarded the consensus national title, even though they lost two games themselves. They played against a one loss Ohio State team in the National Title Game, winning 38-24. But Anderson & Hester gave their National Title to the Tigers.
The Tigers don't claim either of their National Titles, but their SEC moving buddies, the Texas A&M Aggies, did. While the Aggies won a consensus National Title in 1939 they also claim National Titles in 1919 and 1927. They also have one from 1917 that they do not claim.
The circumstances surrounding the 1919 championship are interesting. They were not scored on all year, finishing 10-0. The Billingsley Report and National Championship Foundation were among ten selectors who gave them the National Title. Notre Dame and Harvard were retroactively awarded National Titles for that year as well. Interestingly enough the 1917 team finished 8-0 and was not scored on as well, and was awarded National Championships by 1st-N-Goal and James Howell. They do not claim this championship despite having very similar circumstances just two years later that they do claim a National Title for.
In 1927 the team finished 8-0-1, with a tie against TCU. The Aggies were awarded National Championships by the Sagarin Rating and Sagarin ELO-Chess. It also bears mentioning that there were no bowl games at this point in time, making it harder to decide a true National Champion.
That's a very small look into a very big picture here. One and two National Championships are dwarfed by some other schools. Notre Dame leads college football with 11 unclaimed National Titles, perfect symmetry with their 11 claimed titles. We would talk about them, but this article is simply not long enough to explore all of those in detail. Oklahoma, the proud owner of seven consensus titles, also has ten they don't claim. After that Ohio State with seven and Nebraska and Southern California with six round out the top five of unclaimed championships.
So it begs the question, why not claim a National Title? Perhaps the school doesn't want to look overly concerned with its past or show some sort of respect for “the system”, whatever that is. Perhaps the syndrome of 2003 is still alive for a lot of folks. That year, if you recall, LSU won the BCS National Title, but USC was #1 in the AP poll. However, the Trojans were left out of the Sugar Bowl (the National Title Game that season) and had to settle for a Rose Bowl win over Michigan. Perhaps the desire for one National Champion per season is desired amongst the schools.
However, back to the examples above, The Missouri Tigers should claim 1960 as a National Title. They were ranked #1 in the country before losing to the Jayhawks. The conference says they won, so you're the highest ranked team that won every game. If that's not worthy of being claimed as a National Title, tell what is.
Now, 2007 is a bit more iffy. One team had their number that year, and those Sooners won 11 games. However, they got routed by West Virginia in the Fiesta Bowl, a Mountaineers team that also barely missed out on playing for the National Title after losing their season finale to Pittsburgh. LSU had two losses themselves. They lost to Kentucky, who finished 7-5, and Arkansas, who finished 8-4. So Missouri lost to a better team than either of the ones LSU beat. That might be their lone rationale for even thinking of claiming that title.
For the 1917 Aggies, with such similar circumstances to a National Title they claimed two years later, why not claim that title too? It just makes sense.
But should all the schools listed claim these National Titles?
Why not? If someone offered you $1,000, would you not take it because someone else might have also gotten it or be more deserving? That is doubtful. These teams put in the work and had great seasons in those respective years. The fact there was no organized way of awarding a champion should work in the favor of the schools. Someone thought you were the best, and it doesn't matter who.
Take pride in it and claim those National Titles.