How to grant access to edit/create user-defined fields without full Customizer role

We need to allow users to create attributes and add them the to screens that use the USER-DEFINED FIELDS tab, but don’t want to give them the Customizer role.

They need to be able to get to here:

And it looks like you should be able to grant access to this screen to solve the issue:

But adding just that functionality doesn’t work. The user still is not able to manage user-defined fields:

What am I missing?

Thanks!