No, unfortunately, we don't. In fact, the US is, for all intents and purposes, a country without a functioning public education system, especially at the primary and secondary level. The college picture is a little brighter, but not much. As for your question about "illegals", let me remind you that we currently have an undocumented population of around 12 MILLION people who are already here, and most of them have kids in US schools. We manage okay, but just barely.
Do we have too many teachers? The answer to that is most definitely NO. There's no money in education unless you're teaching and doing research at the college or university level, and even there, people have to fight for money and resources to be able to do their jobs effectively. Teachers in America have lives that are regulated by a lot of different people- parents, the principals at their schools, the districts they teach in, the state legislatures, the UNIONS that they join, and on and on. Being accountable to all these different stakeholders means that the teachers have precious little time to do anything on their own except follow a preformulated schedule that is set by the district. They "teach to the tests" because they have little choice if they want to keep their jobs. And that's wrong, because it interferes with real learning on the part of the kids in their classes. As long as we continue to use the current educational model, which places so much value on grades and standardized test scores as the prime indicators of student achievement and teacher performance, things aren't going to improve.