I find it completely strange that dental care isn't just considered part of standard healthcare. Like, so my employer's health care plan covers every part of my body except my mouth? Why does my mouth specifically need its own plan?

It's a really unfortunate historical accident, especially in the US.

Dentistry evolved relatively late compared to regular medicine, and early oral procedures were mostly tooth extractions, which ended up being predominantly done by barbers, who would also do surgery (!). These procedures were often considered crude and beneath that of a trained doctor, and they were generally performed by self-trained practitioners. There were several attempts in the 1800s to integrate dentistry into mainstream medicine, but they failed, both because the doctors of the time didn't think of dentistry as being a real science, but also because, as dentistry started to legitimize itself, the dentists themselves preferred being separate.

For some reason the same separation also evolved in the U.K., but it's more integrated in other countries. For example, basic dental coverage is part of national healthcare in Germany and Japan. In the U.S., dentists have their own schools and licensing boards and so on, which isn't the case in the rest of the world, where dentistry is usually accepted as a regular branch of medicine and taught at the same universities.

Nor your eyes for some reason either.

Here's an article about the history of that:

https://www.nbcnews.com/better/health/reason-your-dental-wor...