With a dingle element drogue I havent noticed a big shock loading on deployment. I ease it off the stern with a lot of wraps around a primary winch
and continue to ease it all the way out. I used a very stretchy dynamic 11mm climbing rope
once on one. Big disaster. The stretch (and too short a length, 50m or so) enabled the boat to surge against the drogue, which popped it out of the water
and launched it forwards (chain and all) out of the water
about 15 meters. Then we would start surfing and the snatch as the drogue bit again was brutal. The snatch would cause the drogue to pop out again... repeat until we added 100m of warp to the mix in parallel.
Now I dont like stretchy drogue lines. The give should be in the drogue as it slips through the water, not the line. Id be happy enough with a dyneema