I've always had a fan theory that navigating the mycellial network requires that the navigator be a conscious entity for one reason or another. 

Having a conscious navigator is highly beneficial, but isn't a necessity. Discovery was able to perform spore jumps prior to that, it's just that their range was limited, as their computers were not powerful enough to make the calculations required for longer jumps.

Starfleet seems opposed to the idea of a sentient AI in any era, so that leaves Zora as the only "computer" that could operate the spore drive without a flesh-and-blood navigator.

I wonder if Zora could. They may be conscious, but they are still a computer, and might lack the intuition required for mycelial travel by virtue of not being biological.

That would explain some of why the Discovery is still in service.

It would. But as someone else pointed out, Discovery is currently housing several time-displaced humanoids from the 23rd century, and a sapient comouter. Evicting them might be more unethical than letting them continue piloting a comparatively ancient starship, especially if they can refit it to something close enough to modern starship performance.

It might no longer be the bleeding edge starship that the Crossfield used to be, but it also doesn't need to be.

Also, we've seen on screen that the navigator must have some kind of connection to nature and/or the fabric of the universe itself.

I'd disagree there, and say that it seems it be more due to needing a way to commune with the spores to make it work.

Humans can't do that, so Stamets is only able to do that using the Tardigrade DNA, mimicking the connection that it had with the mycelial spores. By comparison, Book could tap into his empathic abilities to the same end, and wouldn't need that kind of modification.

Although it does raise the question of how much of the Mycelial network navigation is due to the navigator, how much is due to the drive itself, and how much is due to the spores/network itself doing things in response to the intent communicated by the navigator.

Logically, any other species that could communicate with the spores in one way or another would also be able to take starships on the same journey, even if their way of communicating with the spores is different to that of Book or Stamets.