I think [new-gm] should be renamed
Specifically to learning-to-gm, but before voting, read on (or skip to the end) for why I think it fixes the problems of new-gm.
Identifying the problem with [new-gm]
As long as we still have new-gm, it will get misused. And for some reason it's particularly hard to tell when it's misused, and since doubt leads to inaction, it will slowly spread and get us right back where we started. The tag needs a new name that leaves no doubt about whether it belongs on a given question or not.
An ideal tag is also one that a new user can use correctly every time; less ideal, but more practical to achieve, is a tag that gets used by new users correctly most of the time, and is clear to us when it's not.
new-gm is not even close to being usable correctly most of the time, and it's not obvious when it's misused. Keeping it and assigning it to one of the categories for correct usage will actually make that problem more acute. As well as using better tags for other categories now covered by new-gm, we should kill the new-gm tag by synonyming it to a new, better name that describes its category better, so it will stick out as misplaced when it's put on the wrong question because someone used new-gm and it was synonym-replaced.
Exploiting synonym replacement for easier cleanup
For an example, if we were going to use [gm-cultivation] as the replacement (not a great tag, just an example for sake of discussion, as it fits a "making new GMs" category of questions), it would apply to questions about teaching new GMs, how to encourage people to take up GMing, and the like. Someone who put new-gm on a question about "How do I write my first adventure?" would actually end up putting [gm-cultivation] on it instead—obviously wrong, and the lack of doubt making it easily fixed.
We should exploit that effect of synonyming to a more precise tag, to make cleanup easier after a question is posted. The rest of the tags to cover the categories now covered by new-gm should be similarly precise, so that they only ever get used when they really fit, instead of just when someone flails around typing in "GM..." and looking for some kind of "I am the [gm]" tag.
A bunch of tag ideas
What follows isn't a proposal, that's farther below; this is just showing my work, so if my specific proposal isn't popular, maybe the process will inspire better ideas.
The principle I focused on while brainstorming: Ideally, newness should be relegated to a detail in the question, with the actual tags describing something more concrete, like a goal, action, or process that's specific to being new but isn't newness itself. As newness is a subject judgement of quality, we can't judge when newness wears off, and tags work better when they represent something objective in the problem-question that can be pointed to clearly.
So some tag ideas, some overlapping and not necessarily dividing along the categories doppelgreener outlined, some too specific or too broad to be useful, some less serious than others, but presented to spark discussion around how to replace new-gm:
Most of these are admittedly not very good, but, well, that's brainstorming. We only need one or two gems to improve on new-gm, anyway.
A lot of these are redundant with gm-techniques. That's on purpose: right now, gm-techniques is a generically-named umbrella tag for many specific techniques, but maybe some of those techniques are actually widely-asked enough that they merit their own tags? We already have some tags like that, so there's precedent too: for one example, encounter-design is clearly a GMing skill, but a big enough subject to merit its own tag. As a bonus, some of these decouple a supposed "GM technique" from the GM: system-introduction is likewise a GM technique that's big enough to not be hidden under the gm-techniques umbrella, and bonus, it's not limited to GMs by the tag just like how introducing people to a system isn't necessarily done by a GM.
Thoughts on that list
The best tag in there, I think. It describes all but one or two questions in the list of "gold" that I identified in my answer to "burninate new-gm tag". It has, I think, a pretty clear division from gm-techniques, in that once you're asking about GM techniques, you're no longer asking the puppy-stage "GMing, how do I even?" questions anymore, so [learning-to-gm] and gm-techniques should be exclusive of each other. (Basically, if someone puts [learning-to-gm] on a question, but to accurately describe the question it should really be [learning-to-gm-better], then it's a mistag and should probably be replaced with gm-techniques.) It also, with a bit of a stretch, covers teaching GMing, in that the asker's target is learning-to-GM; then maybe [teaching-gming] and/or [gm-mentoring] could be useful synonyms.
Similar to [learning-to-gm], it covers a bunch of our questions (not those I've highlighted in the "gold" list, but still some of the best in new-gm currently). It would be very useful for us for finding all those diversely-titled and -tagged questions about "I'm new to this, what the heck?" when we need to find the right duplicate for the inevitable re-asks. It would probably get self-tagged on new questions often enough even by the super-novice users who'd ask such questions. And best, it really narrowly fits a category sketched out by the existence of those questions, which otherwise don't really have great tags on them right now. It wouldn't be a widely-applied tag, but it's nicely precise and good for searchability.
I'm not sure this actually solves much of our new-gm tagging issues, but it's a tag that I think we could use. There are a number of questions that are "How do I talk to the GM/players about not liking the game/doing something/etc." that I notice in new-gm right now that are fundamentally just "how to communicate". That suggests that a "how to communicate" tag would be of use. Also, some of our [problem-*] questions might fall under it, chipping away at another set of problematic tags.
This one has grown on me since doppelgreener suggested it in burninate new-gm tag. It is the topic of many of the questions, after all. I think it has one of the problems that new-gm does though: where is the line between a fundamental and something in gm-techniques? We'd likely see it over-applied, and doubt would prevent tag cleanup, leading to scope creep and back to square one. I still like it though, I'm just not sure how to make it work.
We have a number of questions about GM rotation or making the transition between an old and new GM within a continuous campaign or group, and I've never been happy with our tagging options for them. They're not really group-dynamics as that covers group interpersonal dynamics, they're not problem-gm even when the reason is a problem GM (because that's just question motive, not question subject), they're usually not gm-techniques as it's rarely really about the individual GM's skills. social is almost a uselessly broad tag, but I guess would describe these questions. These questions are specifically about how to swap GMs or generally about how to organise and manage the gaming group, which are subjects we don't really have any fitting tags for.
It's such a small category of questions that I'm lukewarm about bothering with a tag for them, but it's a loose thread that jumped out at me again as I reviewed new-gm questions while brainstorming.
Though I think this tag would fit some questions super-well, I also think it would get way overused and also tread on the toes of terminology fairly often.
We should kill new-gm by replacing it and making it a synonym. I think learning-to-gm is a suitable replacement, because it describes a concrete process rather than a subjective quality, and has a sharp boundary with gm-techniques.
This doesn't solve the swapping-GMs or learning-RPGs question tagging, but it covers and collects together an awful lot of the remaining "gold" new-gm questions, and the rest can mostly scatter to other tags like game-recommendation, system-introduction, sometimes gm-techniques, or just a question's existing tags minus new-gm. Swapping-GMs and learning-RPGs questions are a small enough problem that I think that letting them be a bit muddy for a while longer won't hurt (and what to do with them will probably be clearer after we've finished dealing with new-gm anyway).