Site to site transport is effectively twice the number of transports. People are doing jobs. It might not be the best for them to immediately be beamed to where they need to go, because they might be doing something important. They're generally transporting a few stationary people one place. Their capacity could easily get overloaded trying to get a bunch of excited, moving people to their stations. If they're having to do it in shifts, and waiting for confirmation from each person that they're ready to go, I could imagine it simply being faster and more efficient to walk or use turbo lifts. Plus, that leaves transporters free for emergencies that may arise, or crew members who really can't get to their post quickly.

As far as medical issues go, I can see a whole lot of reasons it might not be safe to move an injured person, and you also don't want a doctor blindly transporting into an area that might have a danger that just injured someone.

Even just from a safety standpoint, you've just identified that some shit is going down. Why put people in a transporter buffer, where a power surge or failure would mean mass casualties. It's a good standard procedure even if it's only meant to not incentivize an immediate attack specifically designed to kill a massive amount of people transporting around the ship. Why distract the crew members doing important tasks with dozens of people materializing around them? With all the movement on the ship, i'd think it would take exponentially more capacity just to track the surroundings and make sure the area where someone is materializing is empty. I assume the transporter is sophisticated enough to handle all this, but it has to have an impact on capacity.