Lets get to the simpler answer first, browser support for variable fonts: Firefox developer editions have some level of support, now Chrome 62, Chrome Android, Safari iOS, and Safari. Based on usage, Chrome + mobiles & Safari hit the majority of the most used browsers looking at Can I Use. And now that Chrome has hit general release, the other browsers that want to stay in the race will be following suit.
As with everything browser related there may be some different implementations and some commands supported or not supported. Best to check the release notes for which ones you expect to support and use settings common to all.
Workaround: you can use CSS attributes instead. Not as fancy, but they are cross-browser compatible and they work. You can set a default in your CSS and then modify the font to variable settings in the places where you want variable fonts. Try a short bit of CSS on a couple of browsers and see if this gets the look you want. I think setting it in CSS is safer than allowing the browser to use it's default -- by user that could be almost anything.