I may be wading into a minefield here

You are. And not only that, you're in a community aimed at trans folk, offering advice to trans folk who are aware of the reality of the situation in a way you very likely are not. That's not a good position to start from when it comes to offering advice. Doubly so, when your advice appears to be coming from a position of "in a perfect world" rather than from the practical realities that trans folk have to deal with in the world as it is.

I've learned it's actually pretty important for doctors to know someone's status as trans

As with all things, this depends on the context. Sometimes, it's relevant. Mostly, it's irrelevant. And sometimes, knowing causes doctors to make mistakes about our healthcare needs when they incorrectly assume our medical symptoms align with those of cis folk of our assigned genders (this is particularly likely if the doctor is not familiar with trans health care).

On top of that, there is a thing called "trans broken arm syndrome", in which doctors tend to immediately aim for HRT or transition surgeries as the cause of whatever ailment the trans person has. Again, this is particularly true with doctors that don't often treat trans patients, or worse, that hold anti trans opinions (even if they keep those opinions private).

Yet, even when it is relevant, telling the doctor can lead to all sorts of othering and exclusion. Sometimes, it's outright transphobia and misgendering. Sometimes, it's being isolated from other patients, because the medical staff don't know how to deal with you. Sometimes, it's just medical curiosity, where the doctor just wants to ask all sorts of irrelevant questions out of medical/personal curiosity, because they don't often deal with trans patients.

I live in a very trans inclusive country, with protective laws. I'm openly trans, and wear a trans flag dog tag, and a trans flag arm band. Yet one of the few situations where I won't openly out myself unless I have no other choice, is when dealing with medical staff who don't recognise those flags. And I do that, because the folk who don't recognise the flags are the folk more likely to other me, more likely to be confused by me, and more likely to ensure that my interaction with them is as uncomfortable as possible. And that's in a safe, accepting country. Imagine what it's like for folk who live in places where transphobia is not only common, but sometimes legally mandated...

Which is to say, each and every trans person navigating healthcare has to decide for themselves how to walk this tightrope. And general advice of "you should tell them" suggests you're not familiar with the lived realities of trans folk, despite working in a hospital