I tend to agree, and would further agree that the dice rolling orgy that is initiative is a bit silly sometimes. I see a few options, if we assume that you're getting rid of individual initiative.
(1) Roll initiative once in the first round. This is simple and fast. The tradeoff is that the fight is very predictable, with no chance for one side to "seize a tempo" and go twice in a row.(2) Roll initiative every single round. This can get complicated. I'll admit, I've forgotten to check initiative in the quick "they go/we go" tempo of an OD&D style game. It also results in fairly frequent changes in tempo, but they're totally random.
I don't really find either of those satisfying, although I'll admit that I find the former to be better if for no other reason than it is simpler and faster. If the players are not making any significant choices and the outcomes are totally random, I tend to prefer quick, fast solutions.
While taking a firearms class recently, I thought about seizing the initiative in a fight. There are a few thoughts here. First, acting is better than reacting. When you act you often get a small window that can be exploited through speed, surprise, and violence of action. The Tueller Drill is perhaps an example of this; this drill, named for a police officer, shows that a melee attacker can cover about 7 yards in the time it takes a defender to react, draw a sidearm, and get a shot on target.
In personal experience I've found that getting that optimal reaction time is highly dependent on a vigilant state of alertness. If you're dialed in on the potential threat and postured to react then the reaction is much faster.
I've also found that there is a significant and adjustable tradeoff between speed of response (or action) and the accuracy of that response. For example, I've done some competitive shooting. There is a balance between speed, accuracy, and power. I can get bullseyes at relatively far ranges fairly easily with a low powered weapon and plenty of time. As I push myself to go faster and faster, using a combat-caliber weapon, accuracy degrades. But, accuracy may still be "good enough." For example, in some competition disciplines you get full points for any hit in the vital area or on a steel plate -- it doesn't have to be a bullseye. In this type of competition it is better to go fast and compromise accuracy as long as you're still in the target area.
The question is how to capture this with an efficient, elegant mechanic.