Agreed with all the above really.
It would be a small Quality of Life improvement to warriors. My question is, do assassins also suffer these problems? (Not being able to attack after casting agil/a.speed buffs?) I'm guessing not because their primary attack is just a normal attack not a spell, but warriors rely solely on their spells for damage. I'll be honest, you should also be able to LionRoar a tao's pets and at the same time be able to charge a TDB and if able, connect a hit. Currently this is not possible due to the global cooldown. You are forced to wait 2seconds before casting TDB.
In those 2 seconds the tao has already puri'd his deva, making it a pretty exhausting fight for the warrior. Another scenario is both a warr/sin fighting each other buff themselves, then immediately go to attack each other. Due to the delay a warrior is put at a disadvantage because he cannot cast his offensive spells briefly, whilst the sin can autoattack freely.
I'm fairly certain Euro mir didn't have this global cooldown between LionRoar and other offensive spells, but then it was a fairly long time ago. Might be worth testing to see if it would cause any problems first, then give it a temporary implementation and see how it goes?