Yes he could have. Kirk, if he had lived, would be 130 years old when Capt. Picard took Official Command of the USS Enterprise on Star Date 41148. McCoy was 137 when he toured the Enterprise D. So yes. It could and may still yet happen. The time line is malleable. There are talks about Shatner reprising his role on Picard.
The Com time's out, but say Picard, for example, were to say, "Picard to Enterprise" without touching the Com, it will still work, just like how the Captain gives orders to the Computer.
Pressing the Com is more for making the alert noise, to alert the private party an order is coming through, but even with that you do not need to press it again ...
For novels and other non TV or movie materials that give information about the Enterprise-C, see https://memory-beta.fandom.com/wiki/USS_Enterprise_(NCC-1701-C) (not sure if the page is complete though)
IMHO opinion Data was insulting Picard. The movie reveals that the Borg Queen had been present on the cube which had assimilated Picard in"The Best of Both Worlds" and had survived somehow.
In the final scene in Enterprise Engineering:
PICARD: Let him go. He's not the one you want.
BORG QUEEN: Are you offering yourself to us?
PICARD: Offering ...
I wouldn't take that statement entirely at face value if I were you. Remember that after the Borg Queen was killed, Data told Picard that he was tempted by her offer for a sum total of 0.68 seconds:
DATA: She brought me closer to humanity than I could have thought possible. And for a time I was tempted by her offer.
PICARD: How long a time?
DATA: Zero point ...