Had the plumber out, expecting to replace the water heater. He took one look at it, and said that some of these heaters have balls in them right where the water comes in and where it goes out. That he would remove the balls and that it would be fine. And sure enough in an hour we had hot water flowing out of the hot taps as fast as cold water will flow.
I’m not really sure what he meant by balls in the water heater, I’m thinking he meant filter screens on the intake and outtake. Or possibly one way flow valve. Either way, removing these balls freed up the flow.
I was just glad he only charged my $200 (he did put in new flex pipe) and didn’t sell me a new water heater which would have been $1600. Although because the heater is 11 years old, I’m sure I’ll need a new one before too long. (the plumber did say the heater seemed to be in pretty good shape.)