unfettered ready room access

[Tenacity]

If you weren't already on the "visitors list" would Picard's doorbell even make a sound?

 

[tharpdevenport]

Not unless it scans your finger print, which by the 24th century would be a far out-of-date method of proving one's identity, that would so easily be defeated you'd think Wesley had aged to a kid again and once again showed up older more experienced officers.

 

[Idran]

If you weren't already on the "visitors list" would Picard's doorbell even make a sound?

That almost sounds like a koan.

 

[Greysun]

The point is a non sequitur though, since doors to any room on the ship that wasn't a common area room always had their doorbells pressed.

That was actually my point. I don't think it is a non sequitur. The OP postulates unfettered access to Picard. In fact, people had unfettered access to Kirk. You have to ring the bell or deal with a locked door (or Riker) when it comes to Picard, you can just walk in and wake Kirk up in the middle of the night if you feel like it.
Also, as Riker informed Shelby, it is out of line to take issues directly to the Captain. Riker told her she needed to go through him and if she didn't there would be consequences. I don't recall Spock ever making that distinction with a crewman. There's also occasions where characters asked to see Picard and Riker told them he was busy (the two that come immediately to mind are the Romulan defectors).

IT doesn't necessarily reflect him being busy or not being busy.

I think that if Picard is in a meeting, it's quite plausible for someone to come to the ready room door, ring the bell and not be granted access. The whole bridge crew is right there, I think someone would say "He's in a meeting with Admiral So-and-So. You'll have to come back later." Picard could also ask not to be disturbed because he was busy which would result in someone being turned around by bridge personnel if they came up there to bug him.