FWIW, a potentially straight-forward solution to this is to get an existing embedded language, such as PHP or Ruby’s ERB, duplicate its configuration file and then changing the embedded language to Elixir.

I did this on Sublime Text, copying from Ruby’s ERB, and it was relatively easy and it works quite well.

I posted an update about the configuration fo beautify to format EEx template as good as possible now. Works quite well now for me.
Have a look at https://github.com/fr1zle/vscode-elixir/issues/129

In settings.json, instead of setting "files.associations": { *.html.EEx: ... } to "HTML", set it to "HTML (EEx)":

"files.associations": {
    "*.js": "javascript",
    "*.ex": "elixir",
    "*.exs": "elixir",
    "*.eex": "EEx",
    "*.html.EEx": "HTML (EEx)"
},

I just stumbled onto this, and @tme_317 has already found it, but you don’t need Beautify.

Once files.associations for .html.eex are set to “HTML (EEx)” it appears that .eex work as an embedded language inside of .html, so they both work seamlessly. Snippets extensions work, .eex tags autocomplete, intellisense and syntax highlighting work, it’s all back.