As someone who is not LGBTQ+ but is trying to understand it myself, the biggest key to "getting it" is separating sex and gender. Sex would be your biological make up, regardless of genital configuration. That's your classic XX vs XY vs the rare "non-standard". It's important to note that there are multiple types of different physical configurations that are "male or female" - Anatomical, Hormonal, Chromosomal, and Cellular - and they frequently do not line up 100%.

Gender is separate from this as a mental construct/thought pattern. For most people, these two things line up, and that is your standard Cis person. A Trans person is where their mental configuration does not match their physical configuration. When this causes significant mental distress we call it gender dysphoria. For the rest, we call it being Trans, non-Binary or something else.

Isn’t that where they would receive the best treatment for themselves? Aren’t gender pronouns useful when distinguishing between biological differences in the medical field?

Why would there be a difference between gendered wards in the quality of the care? Health care is tailored to the patient, where in the hospital they are located shouldn't affect it.