Yes, when I mentioned "connect one layer to the center pin, and the other to the ground, as in the example attached" I forgot to say that you have to make also the thick ground plane as in the picture.
Otherwise your Antipodal Vivaldi antenna will be connected to the SMA ground through a thin PCB trace, which actually is an inductor.
However, instead making an antipodal Vivaldi, you can make a well known microstrip-fed Vivaldi slot antenna, which may have a bit less bandwidth but gives better VSWR vs frequency.