How use Basic authorization in API Action on frontend

[vv-a2007]

Of course, I started with this, but a very strange problem arose which I described in another post

API Action working on front and not working in Server Action

Sorry, I don’t quite understand where I should and what to check. I have these settings and I’m a little confused by the fact that in the local root folder it is /public - I don’t know - is this how it should be? In any case, it is impossible to change it - the SAVE button does not respond. [local] [prod] [project]