Thanks, Thinman. I'll give that a shot shortly. My thoughts were that if removing everything and going "straight from the source" were too low total volume throughput, no amount of removing stuff down line could help. But maybe thats wrong somehow.
Relevant part of shower, with hand sprayer above:
End of hose that attaches to the wall (couple stacked little rubber rings + a filter):
This thing is in the main stem after I detach the hose. It is removable and seems likely to be a flow limiter (it has a spring involved). When I talked about the output earlier without the hose I meant without this thing, directly from this area into a measuring cup:
The hose into which the sprayer attaches (just a hose from this angle) and the end of the sprayer head itself (the one with the green an the metal circle in it: