The ambitious/arrogant tone of "I"

Short answer:
Sense of self and ownership

In the college essay, you sell your opinions. When you say, "I think that...", "I believe that...", it signifies your conviction about the topic. If you feel that you are using "I" repeatedly that it becomes boring, you can shift with a strong, convincing statement. Right now I'm listening to Ziggy Marley's

Love is my religion.

I think that he was convincing enough with that line alone.

As for the curriculum vitae, I concur with @Bruno's comment. And I would like to add more. A CV is all about selling yourself. Since it's all about yourself, there is no need to say "I" for every accomplishment. Several of the following types of phrases would be very convincing.

Managed a $100,000 project from planning to turnover

Using "I" can be limited to the objective or narrative summary section. Even the narrative summary section can be written in the third person.

EDIT: Strong, convincing statements aren't necessarily wrapped in single sentences. One way I might do it is this. Imagine yourself being in the Jedi Council debating whether Anakin should be trained:

Yes, he may bring balance to the Force. He may have high levels of midi-chlorians. The Force is strong in him. But does that necessarily add up that he should be trained?

No.

Then follow it up with your arguments,

I sense much fear in him. Fear leads to anger. Anger leads to...

You probably know the drill :). There are many ways of getting your point across convincingly. It just takes practice and lots and lots of reading.

I have read hundreds of CVs in my past position, the thing was looking for was precisely what the applicant had done. If they said "My team did X" the I pretty much assumed that the applicant had nothing to do with it. If you have done X, then you should state this in an unambiguous manner. That being said, "I was the best manager in history" may sound a tad arrogant whereas "Under my management, team output was increased %137" is just stating a fact.

I share your concerns about the use of 'I' in this context. As a workaround I use bullet points as this form allows me to write:

ACME Industries 1989-2001.