Well to start with, I don't support "spore drive" at all since it seems like Gaea-worship claptrap and violates the idea that no natural phenomenon can exceed the speed of light. And I thought having parts of the Discovery rotate in opposite directions didn't make sense since it wasn't designed to have "spore drive" to begin with so why would it? As for transporting though, I think the main issue there might be that in the one TOS episode where it was mentioned - Day Of The Dove - the issue seemed to be that they were at warp. When the ship isn't "moving" it doesn't seem like it should be difficult. If anything, beaming down to a planet while orbiting way up etc, seems more problematic. How could "pinpoint accuracy" be harder when you're going from place to place within the same moving structure, versus up or down to or from a separate planet or ship?

Teleporting within the ship means no 'crazed person/alien takes hostages' episodes. The original series had one or two episodes where Scotty had to find a way to open the door to the engine room. If crew can teleport anywhere you can't have locked down areas. If the engine room or other level of the ship is attacked and there is no means of escape.... just teleport!This lessens the drama.

If the Klingons captured The Discovery you could end up with a hostage type scenario but the Klingons would have to shut off the internal teleport function otherwise the crew could escape too easily!

There are always outs. Transporting to Engineering could be verboten because of all the energy stuff there, and/or maybe shielding to protect the nearby areas of the ship... in other situations, they could just wave the magic wand and set up a force field wherever happens to be convenient. Evidently those ships have force field emitters EVERYWHERE. Plus in many situations they were at warp as mentioned before... for The Naked Time they could say power levels were too low (although they didn't know Riley had shut down the engines until they got inside) or the increasing gravitic interference from the collapsing planet made it too risky...

"Spore Drive," like the technology that allowed a star ship to hide inside a planet, may be "banned by treaty" or mutual (closed door) agreement. As for transporting between compartments/ship levels, NCC-1031 is/was a scientific research vessel, "with 100 labs" working on who-knows-what. Still, if The Tardigrade was about to break containment, the crew of the U.S.S. Glenn could have "dumped it in space," given the opportunity. (Which they clearly didn't have.) For the moment, I'll go with the "ship full of Edisons" idea, since The Tardigade wasn't "beamed into space," but ejected from a containment tube.

Seems to me the "explanation" that Michael gave for its dna/structure in episode 4, I think it was, meant it "had to be" some kind of genetic mutation from Earth. But I suppose they might have contradicted that later, wouldn't be the first time.

And if it's supposed to be a space creature, then ejecting it into space wouldn't kill it.