People have beamed into and out of sealed caves, buildings and other starships
Quite true - but it
is a matter of penetration depth. Walls of stone up to two kilometers thick are no obstacle, but that's still "line of sight" in the sense of "never over the horizon", which is fairly common usage.
The communicators have no trouble reaching over the horizon—or do you suppose Federation starships deploy communication relays into orbit every time they approach a planet?
Relays would solve all the problems, but we never hear of anything of the sort. Then again, radio waves and Class M atmospheres would be a working combination for curving/bouncing communications even today. Transporter beams are a bit less likely to bounce because of the very penetration that we already discussed!
The "standard orbit" as synchronous doesn't wash, either. There are numerous episodes establishing orbits as much closer.
There is no height limit for synchronous orbit as such - merely for
freefall synchronous orbit. And TNG frequently mentions synchronous orbits above places other than the equator (say, synchronous orbit above the south pole in "Power Play"), which categorically rules out freefall orbiting. Starship engines should handle hovering over a spot trivially easily, regardless of hovering height (one gee to fight at ground level, something like 0.8 gees to fight at low orbit, still 0.5 gees at the orbital heights we usually witness in Trek).
Assuming powered holding patterns (the thing we call "orbit" when we speak of aircraft) also explains why starships fall from the sky when their engines fail. And why they seem to make absurdly tight turns when the camera follows them through part of an "orbit". They are just doing figure-eights over the landing party!
Lastly, even if we assume Flint's technology is line of sight, all we see is a handheld remote control. His planet may have the necessary hardware arrayed to cover all directions.
And he won't even need that if
Kirk's technology is line of sight, because the starship would then be hovering directly above Flint's castle anyway!
(e.g. "Mirror, Mirror" with target cities passing out of range over the horizon.)
A clear case of not hovering - but an interesting contrast is found in "Miri", where the ship approaches the duplicate Earth along a path that would take her past the planet on the right side (the sequence originally filmed for "Where No Man" and seen in the opening credits, with the red planet). Kirk then calls for a "fixed orbit", and we next see the surface of the planet moving from left to right, as if Kirk's ship were flying from right to left at fairly significant, non-freefall speed (plus the planet of course rotating against that movement, since it's a duplicate rather than mirror Earth

, but necessarily at an insignificant speed).
Perhaps the folks in the Mirror Universe just do things differently?
Timo Saloniemi