Yes a Pusik Hitch is configured to set and hold that's all. Once you get into hitches that need to perform as adjustable hitches under load the variables explode.
I don't think what you're describing has anything to do with breaking in the hitch cord (or climbing line).
Important factors to consider for a functional adjustable (under load) hitch system:
a. hitch cord construction and diameter
b. climbing line construction and diameter
Sterling HTP (you mentioned possibly trying that) is not a great choice for the foundation of a multicender climbing system. The Sticht Hitch is acting as a multicender. The HTP core has been behaving badly for many climbers using multicenders. Been a while but you could probably find some discussion about that on the 'Buzz.
As far as the problem with a hitch not grabbing SRT (or SRS depending on what you call it) if there is load on the line from a foot ascender, the hitch may not grab, the hitch needs to be taking all the load to grab consistently.
This is a very subtle aspect of climbing SRT with a hitch and a foot ascender. I you sit back on the hitch during ascent you need to lift your foot slightly so your leg is not loading up the line at the moment you want the hitch to grab. I think experienced climbers do this automatically without realizing what an important moment that is in the climbing motion.
Typically climbers are not sitting back on the hitch during ascent cycles, only when they stop to rest, or for any other reason. If a climber sits back on or loads up the hitch every cycle during ascent they have to be more aware of lifting the their foot ascender up a little so the hitch (if well configured) grabs.
-AJ