We don't know when Picard took command of the old tub. But the longer he spent out there, the more he'd need "downtime" at Earth to forge all the important connections that would give him flagship command. A longterm field commander might be exceptionally unqualified for the job, really, having missed out on all the cocktail parties.
Then again, Picard might have made powerful friends before his Stargazer stint, too. But why would he then be given the rust bucket? Was he sent to the frontier by his handlers so that he would be out of sight and not gather San Francisco muck?
Noncanon material goes with the idea that Picard never even served on any other ship besides the Stargazer, and did that for two decades straight. Canon doesn't claim that, but it does claim that the relic was Picard's first command as a Captain. Most of the stuff at Memory Alpha is bullshit, though: no onscreen source says he was aboard in 2333 or held any particular rank at that time or got the command of the ship straight away after his heroic standing in for the skipper, say.
I used to think there was a canon reference to the Stargazer also having been the first ship Picard served on, but nothing of the sort seems to exist. All we know is that she was the first ship he actually commanded (although with the definer "as a Captain" or "as a captain", so there may have been less formal or temporary stints, or a command while he held Commander rank, or whatever).
Timo Saloniemi