Hmm, I feel they would still need a speed. They don't need to be called "reaction," but I think at least a 1 per action seems appropriate. It also begs the question about extra attacks and action surges. It is a serious nerf for the fighter if it takes 12 seconds (24 w/ action surge) to complete his/her "round." However, it is a serious upgrade if give them extra attacks (like legendary actions) for free.
It seems a +1 in all instances makes sense:
Level 20 Fighter w/ action surge, 8 attacks in: 3+1+1+1+1+1+1+1=10 seconds
Level 20 Wizard casting 9th lvl spell: 1+9 = 10 seconds
Legendary Monster, multiattack + 3 legendary actions: 4(?)+1+1+1 = 7 seconds (still better than the PC)