I have a number of profiles I like to use to help quickly differentiate one tab from another, so I like to be able to quickly and easily choose one when I open a tab. The "Settings -> Switch Profile" menu provides for exactly this but whenever I first start KDE it shows only one. I then have to go to "Settings -> Manage Profiles" and tick the profiles I want in the menu.

That "Configure -- Konsole" window has an "OK" button which applies the selections for the duration of the login, but the "Apply" button is always greyed out, which is I presume why when I restart KDE next time the selections aren't applied and I have to go through it all again.

What's up with that?

Works here (despite the Apply button remains disabled)

You can edit the "Favorites" key in the "[Favorite Profiles]" group in ~/.config/konsolerc (resp. check whether, when and how it changes)