Yes, I think this sort of thing would fit better into Normal Modes than full on hard-enrage timers for encounters.
Correct me if I'm wrong, but I think the purpose of enrage timers is basically to protect players from themselves and doing stupid things like fighting Absolute Virtue from FFXI over 30 hours nonstop, and to prevent cheese compositions like bringing 8 WARs instead of a normal party with jobs that can actually die. Not so much to give difficulty, and for the most part it's fine that the normal modes no longer have hard enrages IMO.
Many fights in this game do have shorter mid-fight dps checks, starting as early as Ifrit Normal's nails, and in fact the ones that don't for trials are sort of the anomalies. Earlier on in the game, more dungeons and raids also had these. We see DT Trial 1 has this while 2 and 3 (and the raids) do not.
Meanwhile, the cool dps checks I can think of are plentiful in Alexander (thought you wouldn't be aware of them if you ran them the first time today, mostly due to players being far too powerful):
a1n kiting and killing adds under missiles before they shrink players and before they impact the arena
a3n every add missed on the whirlpool phase causes damage to the group
a4n not culling adds is a quick way to get too much debuff and explode the raid
a5n more chances to have silly names on adds like Gobbledygroper
a7n without targeted dps cages could kill isolated players
a9n beyond the core mechanic of smelting, faster dps meant you could kill the Jagd Doll in a quadrant that would dip into the slag pool sooner and remove the boss's defensive buff sooner
a10 just has the typical mid-fight dps check with the Gobroller
a11n has E.D.D. an add that blows up if not dealt with reasonably quickly. But, special mention to a11s: E.D.D. has a mid dps check dps check, you have to break his backpack before he finishes a cast or he gets full hp and starts one shotting players.
a12n has the mid-fight dps check (builds LB) and the timegates.
Arguably a6n could go harder on each individual boss in the gauntlet because they're all short dps checks and you could continue from where your group was if you wipe on one of them.
It's also refreshing to see the return of things like ice tombs for failing a mechanic on trial 1; a penalty that dps (and thus all players) can help out with by breaking them, a sort of dps save check. We don't get those too often. I think the last one may have been Memoria Miseria Ex from touching the wall, and that was just a single tap to free a player.