I've given the problem some more thought . . .
All modems (modem/routers) have a high-pass filter before the analogue front end (AFE). At the very minimum that high-pass filter will be two capacitors, one in each leg of the pair. The normal condition, with the modem (modem/router) powered on and the telephone not in use, will be a high impedance condition. When the telephone is in use, it loops the pair and a low impedance condition prevails. That's perfectly acceptable and is part of the design specification.
Now suppose one of those two capacitors that form the high-pass filter is somewhat degraded, is out of specification, is a bit manky. The impedance will not be anywhere near as high as expected by the design. When the telephone is used, the modem's (modem/router's) interface will be shunted far more than that of the design specification and it reacts adversely . . . That adverse reaction could, possibly, be as was described in the opening post.
As to whether that is a sensible proposal?