It is tricky (but possible, with cleverness and a careful schedule) to gain or lose energy this way, but it doesn't matter. If your closest approach is well within the sun's photosphere, it doesn't matter how fast you're going when you get there. So, you can do it with essentially zero delta-v, starting and ending with the same total energy as an object would have co-orbiting with earth, but on an extremely eccentric orbit.
It's not terribly rare (on a geological timeline, at least) for comets to dispose of themselves this way.
Anyway, what is so great about dropping them in the sun? Jupiter swallows comets frequently. Mars is a squalid dump, and so is Venus, at least below the clouds.
The sun is always at one focus of the elliptical orbit. You just can't get the orbit close enough to plasma-brake near perihelion without also pushing your aphelion way out. So you have to aim away from Sol in order to get there at lower energy. Basically, a Voyager probe that stops at the very edge of the gravity well and then plunges straight down. Spiraling down while decelerating is faster, but costs more energy. But as you get closer, you can harvest energy from the solar wind and solar radiation, with solar sails, so the amount of delta-v you have to load onto the launch rocket does not represent your entire delta-v budget.
There are ways to trade off time for delta-v, but at that scale, the ways that really make a difference mean that the person that sets them in motion will be ancient or dead before they finish.
