Once you do that, you open the door to Michigan claiming 22, Notre Dame 21, Ohio State 14, Pittsburgh, 11, LSU 8 and Auburn 5.
It just leads to championship inflation because once you start going outside of the wire services almost every year have several national champions. This championship inflation actually diminishes how dominant that Alabama has been since 1936. While Alabama has been awarded 8 or 9 titles, very few other schools can even claim 2.
Those pre-1936 games shouldn't be forgotten, but they shouldn't be lumped into the current era either. For example, when discussing the NFL, champions from the super bowl era aren't grouped together with the NFL champions of the 1920s.