Did he have a drinking problem back then?
I can't say with absolute certainty, but I think yes is a fairly obvious answer, based on a number of observations I have made.
There can be no doubt that he liked to have a few drinks, even in the early days of Zep, but not sure there was ever a "problem". However, by '75 it's widely accepted that he started dabbling with heroin. By '77, the visual and recorded evidence show that he was clearly in its grip. That seemed to last through the end of the band, at least.
When Page finally emerged for the ARMS shows (83?) at the behest of Clapton and Beck, he was a mess. Watch the footage on YouTube, he can hardly play.
What I suspect happened in and around that time is that he was kicking heroin, and increasing his alcohol intake, as many addicts do (read Clapton's biography, it's basically what he did). Mid 80s (The Firm era) footage & photos seems to back this. He started to look a bit bloated, and he was incoherent when he spoke at times. There is an interview thing he did with Roy Harper that is the clearest example, sad to see. Then you have the Live Aid mess, and the 4th of July sitting in with the Beach Boys, also a mess (I was at that one).
Anyway, this pic looks to be around that time period.