If the shul I was in is at all representative it wasn't paying attention to every word that the chazan was saying.
Looking around during davening over Rosh Hashana and Yom Kippur I noticed 3 different types of people doing 3 different things during chazaras hashatz
1. Learning - Many people come prepared with various seforim and spent much of chazaras hashatz learning. I definitely fall into this category.
2. Daydreaming/sleeping - Some people are simply bored and have nothing to do and therefore daydream or fall asleep during chazaras hashatz
3. Paying close attention to the chazan
I don't know the exact percentages, but in my experience 3 is by far the smallest group, a very small percentage. There are very few people who pay attention to every word of the chazan. Of course, the truth is many people fit more then 1 category. I for example, spend a lot of chazaras hashatz learning but there are times when I do listen to the chazan and participate, it depends on what is being said etc.
The bottom line is that there is a major problem here. Most people simply do not really pay attention to chazaras hashatz on the Yomim Noraim. The chazaras hashatz simply does not speak to them as a religious experience.
In fact, I believe that this is a problem all year round, as well. It is just that chazaras hashatz on a regular weekday and shabbos are relatively short, so you don't see it. However, even on a regular Monday very few people actually pay attention to chazaras hashatz.
The reason is really simple, we just davened and therefore it is hard to see the relevance of chazaras hashatz to us. We don't feel a connection/need to have the chazan repeat what we just said.
I don't know what the answer is but we need to acknowledge that it is a problem.