I believe that's more of a font support thing than a browser thing. I'm pretty sure I've seen Firefox render e.g. "male shrug" correctly on MacOS but they're two separate symbols with the default sans-serif font on Windows. I can see it rendered correctly on Windows on sites that define the font to be something that actually has a glyph of a shrugging man.
On what platform does Firefox render the emoji as more than one glyph? Firefox works for me on Ubuntu, macOS, Windows 10, and Android.
The level of ligatures requested in CSS makes sense for the site-supplied font. (I need to regenerate the fonts due to the table at the end of the article using characters that the subsets don't have glyphs for.) If you block the site-supplied font from loading, the requested level of ligatures may be excessive for your fallback font.