There are many industries which have found value in specific people to do the paperwork and deal with process. I don't think it's needed under 10 or so people if you have a good team, but once you're we'll into the 20+ range, I don't understand why so many devs are so set against giving this type of work to someone else.

It's certainly not what I want to to be doing day-to-day, yet our tools aren't smart enough today for me to do well in a larger org without it.

Yes and no. On the useful side, they own some of the process inherent in large organizations. But they also act as a conduit to push that process to team members.

In practice, our scrum masters have been a place for the PMs, Analysts, and QA leads to go as their positions have been displaced. I would like to see more technical scrum masters that can own technical impediments, but that doesn't seem to be happening yet.

I also think that spreading scrum masters across teams will disconnect them from the specifics of the work and allow them to act more as coaches than non-technical team members.

Yes absolutely, a good one is invaluable at helping the rest of the team be more productive.