You're right, there is no respawn in SP.
Gear selection and respawn in MP are handled by the file description.ext. They are nothing to do with briefing.html.
My suggestion to this type of problem is always the same: use the briefing from the Tutorial Mission and get your mission to work with the wrong briefing. Once you have a working briefing, you can play with the text and so on till you get it the way you need it.