OP - "Besides, the rest of the league might like to play Jeffers more, but they're unlikely to be keen on giving up a draft pick to do it."
Nitpic: Teams don't "give up a draft pick" by signing FAs with a QO attached. They keep their own picks.
The team losing such a FA gets a extra pick inserted either after the 1st round, before the 3rd round, or before the 4th round.
Stupid rule, IYAM because EVERY team gets their regular picks pushed back, equally.
But signing teams don't forfeit a pick.
https://www.cbssports.com/mlb/news/mlb-qualifying-offer-contract-predictions-2026-27-offseason/