Add a UI element that enables template parts to be assigned to areas

[jameskoster]

Now that we have an "area" taxonomy for template parts (#28410) we'll need to update the UI so that users can assign template parts to these areas. This is mentioned briefly in #29148 but I think warrants its own exploration, as it may be possible to set the template part area while editing a regular template and when editing the template part itself in isolation.

[david-szabo97]

Just noting we already have some exploration here as well.
#27875

[jameskoster]

Ah yes, this one:

Let's refine the copy in the PR :)

[jameskoster]

Jeez I've been going in circles this week 😂 Closing this in favor of #29295.