You can still do XSLT in the browser. You can serve arbitrary XML and transform it. As an example, Atom feeds on my website (such as <https://chrismorgan.info/blog/tags/meta/feed.xml>) render just fine in all mainstream browsers, thanks to this processing instruction at the start of the file:
<?xml-stylesheet type="text/xsl" href="/atom.xsl"?>
But working with it is not particularly fun, because XML support in browsers has been only minimally maintained for the last twenty or so years. Error handling is atrocious (e.g. largely not giving you any stack trace or equivalent, or emitting errors only to stdout), documentation is lousy, some features you’d have expected from what the specs say are simply unsupported (and not consistently across engines), and there are behavioural bugs all over the place, e.g. in Firefox loading any of my feeds that also fetch resources from other origins will occasionally just hang, and you’ll have to reload the page to get it to render; and if you reload the page, you’ll have to close and reopen the dev tools for them to continue working.