As of the latest Windows Insider builds, Microsoft Edge now has full support for variable fonts!
Specifically, it supports both the font-variation-settings and font-optical-sizing properties. The font-size, font-weight, and font-stretch properties are wired up to the appropriate variable axes too, reducing the CSS you have to update to switch to a variable font. (Side note: automatic optical size switching and the font-optical-sizing property also works with static font families that have opsz entries in the STAT table, as Sitka does).
Demo and details: https://developer.microsoft.com/en-us/microsoft-edge/testdrive/demos/variable-fonts/