Don't let the HHT be an obstacle. Do it by all means, but file the results away in memory for future reference and comparison. But for now it isn't helping you.

If stropping is making the blade worse, you are doing something wrong*. While speed is beneficial, this is only true if you are doing it right. Do it as slow as you need to to avoid mistakes.

I first saw decent shaves when I stopped obsessing about honing, and started doing 60 laps stropping on plain leather. Although I stropped very slowly and lightly, and the strop I used was short and had low draw, it did give huge improvement.

(*) A possible exception: If you are overhoning and creating a burr, that can give the illusion of sharpness, e.g. passing the HHT. But when you strop, or even shave, the burr can break off revealing a blunt edge.