There's no reason the embed wouldn't work, we do things like that all the time. If you have a web server, you just drop the HTML with a full-page embed into its own directory and you've got the URL of your choosing. You can even embed it with the URL parameter hide to optionally hide some of the other elements like the header, footer, and navbar to make it mesh with the rest of your site better.
Not all browsers like embeds, though, so you do need to be mindful of that. Test it out on a few different browsers before going live with it.
Esri used to use bitly with the custom short URL arcg.is, but I don't know if they still do. It was possible to generate short URLs for your own stuff through it.
- Josh Carlson
Kendall County GIS