How to load default profile type on /profile/uid ?

Problem/Motivation

Consider a site where a user has multiple profile types (as opposed to many profiles of one type)
How can I set which profile type to load on /profile/ ?

I note we have a 'default' flag on a profile, but I now assume after testing this is for use cases where someone has many profiles of the same type?

Steps to reproduce

Install D10 + Profile.
Create Profiles A, B and C.
Create Users A, B and C.
Add fields, and save profiles for each user.
Set a differing profile as default for each user

Now visit /profiles/ for each user - note that the 'default' setting is not respected.

Proposed resolution

Would a field on the user account, selecting a default profile type be something that the view could access with a conditional filter?
Is there any documentation on how the view works? (I note we can't set a page or use /profile/%)