A cappella does mean "in manner of the chapel." But not just any chapel: it refers to the chapel, the Sistine Chapel. This is where Palestrina and many other famous singers and composers of the Middle Ages and Renaissance performed for papal Masses. There is no room for instruments in the chapel, at least not in the small space where the very small choir performs.
It can refer to solo singing as well as choral, because, yes, there are pieces in that repertoire which call for solo music and which would have been performed in the chapel. I'd be much more likely to restrict its usage to religious music than to say it can't refer to solo singing.

I think that gospel music without instrumental accompaniment is beautiful. It can be very freeing for the singers, and no one notices if you go flat just as long as you all go flat (or sharp, as my sight-singing class manages every week). If you're making up your harmonies as you go along, though, instrumental accompaniment of the chordal variety can help if your mind goes blank while searching for a note.