Watched the whole video thinking that it missed the point.
Yes, Vanquish is awesome, and yes, for some, quick movement options and acrobatics make for a better shooter, and yes, cover hasn't been utilized in ways that are very interesting in most games, especially The Order.
BUT... my first introduction to cover shooters were with two games that define it for me, and were wholly and incomprehensibly missing from this video - Full Spectrum Warrior and the Brothers in Arms games.
Cover wasn't some safe haven to regenerate health, it was your only way to avoid getting shot, even once, which would seriously hurt or kill.
The idea was Find em, Fix em, Flank em, Finish em, and a set of mechanics that properly communicated how deadly gun combat is, while providing you some reality based means to overcome that danger.
In those games the enemy would hit the field, you'd make contact (preferrably with you behind cover and the enemy exposed -- the Find em part of the formula), and both enemies and players would then seek cover. But at this point you don't just do the pop-up and aim for the helmet routine - you establish a base of fire to suppress the enemy (light machine guns that excel in saturating an area with fire rather than precision point fire), fixing the enemy in place heads-down behind their cover so you could then move your assault element with some safety into a flanking position that could shoot the enemy where they they were exposed. Similarly, you wanted to move between different cover positions yourself to both avoid and seek flanks against your enemies. And if you had explosives you could literally destroy cars or light cover to remove those positions from the field.
In this model you want enemies that shift positions to compensate for the player, to force players to account for suppression and to seek flanking opportunities. This is the epitome of cover based shooting, and its realistic roots in military shooters. It hasn't fared well because shooters have moved into more linear forward only affairs where you aren't nearly as worried about being flanked, along with AI and mechanics that don't allow meaningful suppression. And yeah, add to this the pinpoint precision grenade spam of COD and its clones to force players into the cover where they can shrug off some bullets with their regenerating health and you've made bullets no longer scary, which removes the need to rely upon cover, but you move further away from tactics meant to avoid taking damage altogether.
Gears 1 excelled at this at high difficulties because it simplified this to an arcade level, and put the player into confined areas where you had to fight and maneuver between cover positions while seeking advantage. Cliffy called it a horizontal platformer for a reason - and those qualities only got diminished as the game blew up in size and scope and focused more on spectacle and shoot-the-glowy parts gameplay.
I'm fine with the idea that cover ruins shooters for some, but without tackling games like Brothers in Arms and Full Spectrum Warrior and others you miss the foundational aspects of these systems and what they do right when designed for it.