The official Treknobabble explanation, seen in the TNG Technical Manual and other official media, is that most of the stuff on the holodeck is holographic and can't be carried outside but some stuff (like food and drink) is replicated matter you can really eat and drink. The "mostly holographic but sometimes it's replicated" explanation also covers the occasional on-screen inconsistency like when someone in an episode threw a snowball out the holodeck doors into the hall, or when someone fell in holodeck mud and left footprints in the corridor as they left.

they frequently make a point that things can’t be carried out of the holodeck

Yeah, in my head that's a "safety protocol" thing too. I know it's not really consistent with anything, but it's just an explanation I like.

Of course, even if very large and with pretty big doors, probably not big enough for any equipment they might need.

In my head there's no reason why it couldn't just be a normal human-sized door like any other door unless they needed it extra big for some reason. Maybe the ships are designed with standard parts, so anything they need to replicate in the holodeck can fit through a standard-size cargo door. But, the explanation is probably really "the show's designer thought that door looked cooler for that room".

why is ‘site to site’ transport the exception rather than the norm? Why do they have to ‘go to the replicator’ to get their food instead of it materializing right on their table?

Yeah, if you really think about it, it all starts to fall apart. My thought was along the lines of "energy budget" and/or "margin for safety". But, like you said, it's a show, I doubt anybody really spent much time justifying it or thinking about it.