What is "embed advanced"? A search turns up nothing. I'd hate to make a custom map style, embed a map with it, and suddenly be charged upwards of $14 per thousand calls. Seems pretty steep regardless of what it is. Mapbox doesn't charge that much for anything on a per-call basis.

iOS11 added a "Desktop view" mode but I am not sure the feature is well known (Android has had the feature forever, so by mobile, you mean iOS I think).

From memory, in Safari, tap share, then scroll across to desktop view. I don't have an iDevice with me at moment, but it should work.

By "mobile" I mean Android. And since Android has many browsers "Desktop view" may or may not exist, and if it does exist and can mean many different things. I don't bother to use it because I never found it to work properly. I haven't tried for a couple of years, though.

Desktop mode in Firefox for Android has always worked very well for me, if it helps.

Tap Share, then "Request Desktop Site"

However I have found that it is quite unreliable at working. Not sure if it's just changing the viewport for CSS, or the user-agent, or what. But it frequently doesn't work.

you can long-touch the refresh button for quick access

It's been there since at least iOS 8.