Flying by Jupiter for gravity assist, and with the Oberth effect bonus, has been used for multiple missions to other outer planets and to comets. And has potential for use even for missions aiming at the Sun and Mercury.
While using Jupiter for reasons of orbital mechanics, what kinds of science about the Jovian planetary system would be most accessible and valuable to perform while such a maneuver is performed? Should for example a mission to Uranus or to the Sun or to a Kuiper Belt object or to a Centaur planetoid have a secondary science mission at all during its Jupiter flyby? I mean like New Horizons tested out its instruments, with some Jovian science as a result. But my point being if this could/should be given a much more prominent role during Jupiter gravity assists.
For example, would it be feasible for a mission with the primary objective to study a far off minor planet, to combine a gravitationally efficient Jupiter flyby with passing very close to a small Jovian moon with some common basic characteristics? A mission to Mercury to on the way study a Mercury sized Galilean moon? A mission to the Sun to study Jupiter's magnetic field? Or are the orbital mechanics and the required associated spacecraft maneuvers, as well as the differences in instrument requirements, too stringent to allow for any substantial Jovian science as a "by the way"? I mean, substantial enough for Jupiter orbiters not having to care about this and that, because all the flyby missions do it already at this crossroad of the Solar system.