Halo is the series that pretty much proved that first person shooters, and later, online shooter multiplayer, could work well on consoles to the point where it becomes “the” popular game. As a result, many games started to borrow elements from Halo, including things that nowadays seem quite standard for a “modern” shooter but was not very usual or even somewhat absurd just a decade or so ago.
Regenerating health is just one example. I think the two-weapon-only-including-sidearm thing is another, and so is the designated grenade throw button. Correct me if I’m wrong.
I know CoD didn’t do any of those things until CoD 2, which I vividly remember thinking “wow they really want to be like Halo” while first playing it.