We give up. We surrender. We're packing it in.
In a few moments, I will be calling the plumber to get our hot water pressure issue fixed.
A little more experimentation gleaned that there's something in the pipe between the kitchen sink and the heater. We can flush it out so it's loose, but we can never completely dislodge it from the system. We'll get it to work great for a few minutes, and then it suddenly looses pressure.
We've replaced valves, replaced washers, snaked the line, backflushed scraped with wire, and we still got nothing.
White flag's a-flyin'.