Yes, but because it only modifies the threshold it can leave you in the same position.On pg. 172, the hunt modifiers table lists examples of things the players can do to increase the hunt threshold (travel under false names, take seldom used paths, etc.), so I don't think you need to create houserules in order for heroes to take steps to avoid the Eye.
I should pause to say that I'm not grousing about the official implementation (I save that for Hope 2.0); I just love brainstorming about game mechanics. I think the Eye of Mordor rules are ok. Sure, there's the unexplained wonkiness of what happens when your starting Awareness is already over the threshold, but as has been discussed that just means that the heroes will be harried at every step.
And, likewise, although I can wish that the rules created more opportunities for decision-making, they're fine. They'll work. It's just that none of the three sources for increasing Eye Awareness are in control of the players. Assuming they aren't committing misdeeds, Shadow is (in my experience) usually gained as the result of things beyond the players' control. E.g., you stumble upon a grisly scene, and the LM tells you to roll dice. You have a Journey event, which states that you must pass a skill test or gain Shadow, etc.
Likewise with rolling Eyes on skills, especially with certain LMing styles. You try to sneak past a guard and the LM tells you to make a Stealth test, and if you get an Eye your Eye Awareness increases. Here I'll note that I prefer an LMing (GMing) style in which there are ways to overcome challenges that don't require any rolling at all: dice happen when there is uncertain outcome, and the goal of players should be to minimize their exposure to RNG. And that before dice are called for the LM states what will be required, and what the consequences of failure are. But, again, frequency of rolling varies widely with LM style.
So...following my chain of thought here...maybe a positive feature of this mechanic is that it will encourage players to break the habit of declaring actions with dice. The old, "I'll make an Insight check....18!" Instead, it creates an incentive to think through problems, ask the LM about possible courses of action, and find lower risk alternatives.
Still, I can wish for a somewhat more sophisticated/engaging mechanic that is less passive and more in control of the players. As a simple example, instead of increasing the Threshold by 2 if the company decides to travel incognito, the rules could require the players to actively maintain their anonymity, and while they are successful the odds of gaining Eye Awareness are reduced. And, yes, that is more complicated and TOR (both 1e and 2e) aim for simplicity. Sometimes I hit dead ends with my ideas because I just can't find a clean enough solution.
Again, I just think it's an interesting problem to think about/discuss.