I'm in no way an expert, and, frankly, I casually disregard canon if it offends me. But I can think of a few potential reasons to have a transporter room.

• It removes variables from the transporter process, making it safer and more reliable. You either know exactly where you're sending people from, or receiving them to, so it's one less thing to have to adjust.
• It's a staging area. Even without transporters, today staging areas - where you get everyone together and they mentally prepare to move as a group - are important.
• When receiving, the transporter pad has a lot of extra security options; you can transport things into secured environments. I feel as if they tended to do this more in TOS, but I don't have an example off the top of my head.

I think the main thing is that it's just safer, because one end of the transfer is a fixed, known constant. You can beam people directly to med bay, but you're adding variables and risk, so you only do it in emergencies.

I believe the in-lore explanation is that the teleporter always has to be in the path of any transport. So if going from the bridge to a planet, the teleporter actually teleports you twice: once from the bridge to the teleporter buffer, and next from the buffer to the planet. The room was where the teleporter was physically located.

With improved technology later in the timeline (Discovery), they did in fact abandon the need for the teleporter room altogether.

For what it's worth, they never did address the most fascinating aspect of teleporters: that in the future they solved the problem of how to transfer consciousness. Though the existence of Thomas Riker does raise issues that are unresolved unless you accept that either teleporters do in fact kill you or consciousness can be copied. Based on how willing people are to step into them, you would imagine it's not the former.

Probably energy efficiency reasons.

When you're beaming in or out from the transporter room, that's only one "hop". When you beam from elsewhere to elsewhere (bypassing the transporter room; aka a "site to site transport"), you're actually beaming to the transporter room and then back to your destination, so it's two hops; you just don't materialize between the first and second hops.