Specifically........if a run of gas pipe is put together, in this case
roughly 65 feet total, with about 15 fittings along the way, and the
finished piping shows leaks in one or more joints once the line is
pressurized, how is tightening best accomplished?

Thats a darn good question; I don't think there is much you can do except
start over.
This time wrap with yellow teflon tape and put some dope over that.
Shouldn't leak.

If you have trouble with your joints leaking, good luck with unions.