First, yes, 1>&2 is the right thing to do.
Second, the reason your 1>&2 2>errors.txt example doesn’t work is because of the details of exactly what redirection does.
1>&2 means “make filehandle 1 point to wherever filehandle 2 does currently” — i.e. stuff that would have been written to stdout now goes to stderr. 2>errors.txt means “open a filehandle to errors.txt and make filehandle 2 point to it” — i.e. stuff that would have been written to stderr now goes into errors.txt. But filehandle 1 isn’t affected at all, so stuff written to stdout still goes to stderr.
The correct thing to do is 2>errors.txt 1>&2, which will make writes to both stderr and stdout go to errors.txt, because the first operation will be “open errors.txt and make stderr point to it”, and the second operation will be “make stdout point to where stderr is pointing now”.