Yes, I've done the brake stomp thing when needed, however it is rarely needed. I think you may be too far in the water - if the back end of the boat is still floating when you're already on, that's the issue. Floating the back end changes the angle of the boat just enough to cause that issue. The only other possibility is that the strap isn't wound tight enough, yet. When the rig is on the hard, try winding it in more - don't be afraid to wind hard - with a newer setup, it often takes this extra measure to really tighten the spool up.
I know you mentioned you don't want the bunk slicks, but just for more info... these would allow you to not have to back into the water as much (some ramps can be quite shallow, too) and would make it even easier to slide the boat back on (again, without having to be so deep). Also, with the plastic stuff, you don't run the risk of sand/grit getting into the carpet and scratching the boat.
If you have 2x6 bunks, and you decide to look into slicks more, Tie Down makes a style that is 3-sided - it fits over the bunks like a cap and is secured with screws through the sides of the cap, not the top - meaning there's no chance of a "screw pop" that gouges the boat.
I'm not trying to beat a dead horse with these slicks - if you don't want them, that's totally fine - your boat, your call. But I've been boating my whole life and likely launch and retrieve more boats than the average person because I also work in the industry. Even though you have gotten lots of experience, these only make things better. I even used them with my last a boat, a 28' cruiser!
