Adding prebend opens up a whole range of issues, most predominately, you'll be looking at recutting your main as you've effectively flattened the sail before you leave the dock. If the main is cut with prebend in mind, your fine. General rig tension I'll buy. Shrouds, etc... are the guitar strings you allude to. It deserves a look, but in my experience, prebend is about many things, just not about harmonic vibration.
Mast pumping is about fore/aft mid mast movement that is wave or seaway induced on some rigs depending on the mast section, and as you say,this is controlled with check and/or a baby stay.
Thats entirely INCORRECT.
A 'stock' plain vanilla mainsail will 95% of the time be cut EXPECTING the correct amount of 'prebend' in the mast ... and that small amount of 'prebend' will typically decrease
the amount of maximum draft by about 2-3 inches (as measured at the 'point of maximum draft'. So, in reality, without setting the proper prebend will result in a slightly powered-up
sail with a slightly 'hooked-up' leech.
Setting PROPER prebend increases the relative mast stiffness
by increasing the "I (structural moment of inertia) to the third power" a vitally important characteristic of 'stiffness' / vibrations. The previous posting was about INDUCED HARMONIC VIBRATIONS (caused by the mast shedding 'votices' - Karman Vortices ... not
from wave or sea action - BIG difference.
More applicable info on 'mast vibrations' (although the below is Seldén mast specific the can be applicable in a 'general sense': Seldén Mast AB
Seldén Mast AB
.... but see page 24 for correctly setting up 'pre-bend' on a (Seldén) mast
So the previous recommendation/posting stands, as the OP probably has a thin section mast that is vibrating because its being 'induced' by the wind to oscillate as its 'natural harmonic frequency' and the typical remedy is to change the rig tension AND to add 'pre-bend' to artificaily change that natural frequency.