The curvature of spacetime does wild shit to how you would expect physics to work. If you want to fall into a gravity well, you have to slow down or you'll just slingshot past it.

The problem is, you have so much speed that you keep missing.

The reason you need to slow down is because you're starting on Earth, which means you're moving fast enough parallel to the sun's surface that for every foot you fall downwards toward the sun, the sun's surface curves away by 1 foot. This results in the nearly circular orbit around the sun we exist in.

If you start speeding up, the orbit becomes more elliptical, except your aphelion starts raising away from the sun because now you're moving fast enough that you've moved more than 1 foot sideways in the time you've fallen 1 foot downwards.

Slowing down has the opposite effect. If you get your speed down to 0, you'll fall straight down toward the sun as normal with gravity. But you don't need to go all the way down to 0 velocity to enter the sun, you just need to slow down until your elliptical orbit brushes up against the sun's surface. If you then want to speed back up to avoid falling into the sun, you need to do it parallel to the sun's surface. At this point, speeding up toward the sun will actually make you fall into the sun faster.

So basically the problem isn't that you're moving too fast to fall into the sun. By virtue of Earth's orbit, you're moving too fast in a direction away from hitting the sun's surface.