Ensure CB Core Fields Ajax is public and published in CB > Plugin Management. Once done navigate to CB > Field Management and edit the fields you want to have ajax editing. Next under Integrations > CB Core Fields Ajax under the "Profile View" section set "Display" to "Enable". You should now be able to click on the field when viewing your profile and edit it.

but the edit profile view is terrible

What about it is terrible that you'd like improved? Note you've full control over the layout including multiple layouts you can select from. Simply edit the Default template in CB > Plugin Management then under Layout > Profile Edit you can select from some premade layouts or supply your own substitution supported HTML for a full custom look.

So you're using a custom layout for profile view? Are you substituting in fields with [FIELD_NAME]? If so those can't be ajax edited. Those are raw field displays. You have to use the tab content substitutions explained in the layout parameters or use [cb:userfield field="FIELD_NAME" /] substitutions to ensure the ajax edit HTML can be output.

And for some reason it's now working. I spent ages yesterday but it's just working fine now.

Are you using a heavy amount of caching? It's possible the page was just cached. That'll happen often in dynamic environments.

A small issue is that editing a linked email address opens a new email when editing it. Not a major issue though.

I'm not sure what you mean by this.