The thing is when some people talk about number one centers, they actually are only talking about being a top 5 or top 10 center instead of a top 30 center.
Koivu was a guy who came into the league as a potential elite centre, and he lived up to that briefly in his 2nd season before busting his knee, and it established a certain benchmark that he never got back to.
So then we went through the next decade or so debating whether he is a #1 center but the argument was with wavering goalposts.
To me, he was not an elite center, but he was a top line center.
I think you're right, a good example is the 2002-2003 season (Koivu was 27 years old, right in his prime).
Koivu had 71 pts and a 0.87 PPG, which was good for 12th for centers. Ahead of him were : Jason Allison, Brad Richards, Doug Weight, Vincent Lecavalier (all of which were dubbed as first liners) and HHOFamers : Peter Forsberg, Joe Sakic, Sergei Fedorov, Mario Lemieux, Joe Thornton, Mike Modano and Mats Sundin. Around the same PPG were Brendan Morrison (playing on the first line with right in prime Naslund and Bertuzzi), Andrew Cassels (lol), Robert Lang (Pittsburgh first line center, playing with Jagr or Bondra), Alexei Yashin, Olli Jokinen and Pavel Datsyuk. That's a lot of star power namedropping.
Koivu teammates were : Richard Zednik, Yanic Perreault, Jan Bulis, Oleg Petrov and over the hill Doug Gilmour and Donald Audette.
And he wasn't called a true first liner...