Is the job of the chazan to pray with the community or for them? This is an argument in the final mishnah of tractate Rosh Hashanah. The rabbis say the chazan does not discharge the obligation to pray for a person who knows how to read. Rabban Gamliel says he does, even for those of the community who can read.
Maimonides sees the chazan's repetition as more than a safety net for the less proficient. A chazan leading the community is what defines communal prayer and makes it so more meaningful than praying alone (Hilchot Tefillah 8:1,4).
So important was the communal aspect of prayer, that when people stopped listening to the chazan's repetition, Maimonides cancelled the silent prayer "because of all the men not paying attention to the chazan while he prays the Shemoneh Esrei aloud. People instead would speak to each other and walk out."
Interestingly, the halachah decides in favour of the rabbis' opinion all year round; we are each responsible for our own prayer that we can manage to pray.
But on Rosh Hashanah and Yom Kippur, the chazan fulfils our obligation to pray because the prayers are long and unfamiliar to most of us. So it's extra important to listen to him.