The issues list for the OSM Carto map style is over at https://github.com/gravitystorm/openstreetmap-carto/issues .
You could request there that “name:en” be used when “name” was not available, but I suspect that it’d be rejected because “name:en” isn’t really any more special as far as OSM Carto is concerned than, say, “name:cn”.
A similar-ish issue is https://github.com/gravitystorm/openstreetmap-carto/issues/3870 , which is asking about how to do the same thing in a personal copy of OSM Carto.
There are of course other map styles at openstreetmap.org and elsewhere that show English (or at least Latin) names for everything else. https://www.openstreetmap.org/search?whereami=1&query=8.7683%2C98.3937#map=13/8.7573/98.4039&layers=C is an example - but that style doesn’t show a river name there for some reason.
If you wanted to create your own map style based on OSM Carto then the code change to “show name:en if no name” is trivial - it’s essentially just https://github.com/SomeoneElseOSM/SomeoneElse-style/blob/master/style.lua#L208 . Unfortunately it doesn’t sound like that’s what you want to do.