Victory!
It took several mighty thwacks with a hammer, then prying with two long-stem flat-head screwdrivers, but finally got the plastic cap off the siphon valve. Stripped the cap's internal grippers doing so.
Then discovered the plastic valve itself wouldn't turn, using pipe wrench, vice grips, or channel lock. Yeesh. Had to use a flat head screwdriver as a "convincer" on the internals, which finally loosened it from the brass faucet head. Then used vise grips to turn it out. Not a pretty sight with the top chewed up, but got it out.
Noticed that the small flat rubber valve inside the plastic valve assembly was in its hole cockeyed. Straightened that out, then girded my loins and struggled to get the now-damaged plastic valve back into the faucet. What a merry circus of adventure!
Put the now-damaged plastic cap back on on the now-damaged plastic valve, and re-installed the faucet valve stem, etc. Then reopened the supply line, gingerly turned the faucet handle and ... it works!
(Now ordering complete new siphon valve assembly.)