If you have expertise with a tool, such as Thieves' Tools, it doesnt actually show on your character sheet.
In addition to this displaying on rules-as-written characters, I would like the option to do this in Homebrew.

Indeed! I was looking for this today and agree (and primarily responding in order to be updated if someone in the know responds).

You can already do this right on the sheet. Click on the gear next to the word 'skills' at the bottom of the skill list. The sidebar should open, and at the bottom should be a dropdown labeled 'custom skills.' Click on it, then click on '+Add Custom Skill.' Click on the new skill that appears, then open up the Edit dropdown and set up whatever you like.

This workaround does technically work; but it's inelegant.  I want the option to add the "E" (and a dice rolling option) next to other Tools, in the Proficiencies & Languages section.

I like to keep the Skills section to the primary, base skills, if possible.  Keeps it visually cleaner.

How do you expect that to work with the fact that most tools vary wildly in what stat can be applied to them? An artificer is more likely to use INT for their thieves' tools, for example.

Simply displaying an "E" for expertise shouldn't be impossible.
As for the rolls, allow custom options, like other things.  And if the specific roll is atypical, just do it manually.