Well, this is the premise of “Jump Soles” and as you mentioned the achilles is a prime factor here. Also, just like we weren’t meant to walk with elevated heels we weren’t meant to walk with elevated fore foots either. Best bet (if we’re talking about stiffness) in my opinion is a perfectly flat shoe with cushioning. We are meant to walk barefoot on grass, since many of us can’t do this during the day walking with a flat cushiony shoe is like wearing portable grass, but you aren’t altering your gait as much as an elevated shoe, whether it be heel or fore foot.
This can definitely be hard to find. Flattest shoe I’ve been able to find that is more of a cross trainer/wear it during the day is the adidas samba indoor soccer shoe (for anyone looking, indoor soccer shoes in general happen to be much more flat in comparison to other shoes, just fyi). Perhaps putting an insole can help here. Although, insoles typically have heel lifts too haha damned if you do damned if you don’t type of situation.