That's only the case if you want to go into a controlled orbit very close to the sun. But we don't want to do that.
To crash something into the sun, we'd be happy with any orbit which is elliptical enough such that the perihelion is inside the sun's radius. We don't care what velocity we have at that point, even if it's theoretically high enough to send us back out to the orbit of Earth (or even Neptune) on the other side of the orbit, because the act of hitting the surface of the sun will remove any problems there.
So, we don't actually need to change our speed very much. We "just" need to change direction. Or any combination of direction and speed which gets us within 1 solar radius somehow. As another commenter has noted, we can use slingshots off other bodies in the solar system to change our delta-v in a large number of ways. We should be find a suitable slingshot *somewhere* to get us on a suitable orbit for impacting the sun without needing too much extra propellant.