I don't think they made the later doctors any more important than other crew members, at least not intentionally.
Crusher seemed to have a thing with Picard, and she pooped out Wil Wheaton, but she wasn't more central than anyone else overall. Maybe they did intend to make her that important at the start, but often she was just there.
Bashir was equal to everyone else, as was Phlox (he just usually stood out because he didn't suck). The EMH started out no more important than anyone, but they focused on him more because he was a clear standout character due to Robert Picardo being awesome, which is why they had to invent the mobile emitter in S3.
I think TOS is the only show where the doctor was that important, being part of the "big 3", but the other shows were much more ensemble casts where the doctor had their place.
I think for that kind of show, the doctor role was just important for a lot of the scifi stories they told. It's not a plain old real-life ship, so it's not only about keeping people healthy, they were naturally involved with many plots, like checking over the new aliens and giving information, or making a crew member look like a Klingon, or saving the crew from a disease that turns them all into horny hallucinating senior citizens.