Thanks for replies to my overblown earlier question, and I realize now that what I want is an RJ11 plug strip (or 5 of them) -- in other words, something that looks and functions exactly like a 110 V plug strip, except that the cord on one end has an RJ11 connector on it instead of 110 V plug on it, and the 4 or 6 or 8 110 V receptacles along its length are replaced by 4 or 6 or 8 RJ11 female jacks, all wired in parallel.
Except, looks like *nobody* seems to make such an object, or any low-cost fully pre-assembled functional equivalent to it. Odd . . .