From personal experience: using WebSockets for realtime communication with a microcontroller works really well. I've put together a number of products using ESP8266 to do this and serving out the HTML/JS from an embedded web server. Works particularly well for streaming out the state of sensors on the device.

The main issue I had was that only HTTP was well supported on the Arduino/ESP8266 platform and as we all know that's fallen out of favour. Unfortunately, it also limits what you can do on resource constrained microcontrollers. Having HTTPS in this library opens up the ability to build an SPA which was something I'd always wanted to do.

I don't see any mention of ESP8266 on their site but hopefully, as probably the most widely available IP enabled microcontroller, it will be supported too. Time to do some more digging into getting it running.

edit: I obviously didn't look closely enough - there's recently been added support for ESP8266 - happy days!