HTML is actually a dialect of SGML. XHTML was an attempt to move to an XML-based foundation, but XML's strictness in parsing worked against it, and eventually folks just standardized how HTML parsers should interpret ill-formed HTML instead.

No, HTML was historically supposed to be a subset of SGML; XML is also an application of SGML. XHTML is the XML version of HTML. As of HTML5, HTML is no longer technically SGML or XML.

HTML is far loosier-goosier in its syntax than XML allows. There was an attempt to nail its syntax down in the pre-HTML 5 days; that's XHTML. When HTML 5 pivoted away from that, that spelled the end of these two things ever coming together.

Really, I think you can trace a lot of the "XML is spooky old technology" mindset to the release of HTML 5. That was when XML stopped being directly relevant to the web, though of course it still lives on in many other domains and legacy web apps.