Essentially when you are running and open the tool belt GUI it stops you from sprinting or walking. I often do it for productivity. I am not sure if this is feasible.

I would also like to see this happen, as I had been wondering why the tool belt feels clunky in comparison to other radial menus (e.g. spells in Iron's Spells and Spellbooks) and this is the reason.
Being able to swap tools on the fly while moving around would greatly improve the experience with this mod, hope you can revisit this issue sometime!

This has been implemented in the 1.20.4 version. I have been busy but I want to backport the change to 1.20.1, and possibly 1.19.2.

Oh that's amazing news, thank you! I look forward to trying the 1.20.1 version when you get around to it.

As it is right now, the radial menu is an actual Screen (Gui).
Allowing the menu to work while moving, would require reimplementing the menu as an overlay renderer, but that would mean messing with the mouse input stuffs to be able to "ungrab" the mouse temporarily, and I don't know what side-effects would that cause.
It is an interesting idea though so I'll keep it in mind.

I'd just like to +1 this idea, if it's possible it would be a nice feature.