If you fix the lead / left hip at the start of the backstroke, anchoring it for a reference of how to return square, and the grip makes the putter face aim depend solely upon whether the shoulders' return to parallel / square alignment at impact, then returning the shoulders to square for impact is easy since the lead hip anchor brings them back to the original position at address. This means you can BUST putts straight and square and solid without thought or care at all. Just a firmed up grip and nothing else moving -- no hands, just a shoulders / arms backstroke of any sort that does not go crazy to the inside and don't worry a stitch -- just return / rubberband back to the anchored lead hip and let the "putter face aims wherever shoulders align parallel" do its thing. Turn with the lead hip anchored, re-turn, and that's straight and square and solid without further ado.
It helps if the follow-thru is determined by the shoulders returning to address alignment but not continuing past that, so the thru-stroke feels like a braking of the shoulders returning to square once done that casts the arms and hands and putter head vertically rising down the line without the shadow of any push. So as long as the putter face is firmly gripped so the aim of the face matches / parallels the shoulder alignment, you can actually "go for a push feeling thru impact" but there won't be any at all -- the anchored hip prevents it. So full speed ahead thru impact is okay if you want -- nothing but down the line strokes and rolls. Then settle down to touch but keep the anchored-hip dynamic.