For these reasons i don’t use the GPS overlay of GPS approaches anymore until i understand it better.
Using the OBS mode of the GPS is better for this kind of thing because you set the inbound track (the magenta line bearing) manually.
Just make sure that you set it to the LOC bearing and not the runway bearing because they may not be quite the same. I did once report to ATC that their localiser was off track, not realising it was offset by a few degrees
With the EHSI i use the OBS mode for finding final approaches to VFR fields, but not for the ILS. I mean all you really have to do is to turn a know and set the correct final course and fly towards the LOC with HDG+NAV. All the rest happens automatically …