I always thought (my personal delusions) was that 90% of everything in the studio was his tele and he played his lp live. Imo a tele and lp sound pretty similar in the studio, at least on the vintage recording gear and gain settings of the amps then.
This kind of thing happens all the time with Page - people jump to some extreme conclusions:
To begin with people assume that everything was the Les Paul, then they find out that some of it was a Telecaster, next thing you know some people say that everything was a Telecaster etc etc
AFAIK after the first album the Tele was mostly used for solos & some textured overdubs (NB - "mostly used for solos" does not mean "used for most solos". Two different statements ).
Same thing happened with the Strat - it got out that he used it on The Crunge, next thing, people are saying that Houses Of The Holy is all Strat. Strat on 10 Years Gone? Physical Graffiti must be all Strat... He used it again on For Your Life: next thing people are saying that Presence is all Strat... etc
But until JP spills the beans himself, or JPJ does, what happened with what guitar when, in the studio, is never going to be 100% settled...