LLMs already do quite a few things they were not designed to do.

No; they do exactly what they were designed to do, which is convert words to vectors, do math with them, and convert it back again. That we've find more utility in this use does not change their design.

What if “the internet” developed some form of self-awareness - would we know?

Uh what? Like how would it? This is just technomystical garbage. Enough data in one place and enough CPU in one place doesn't magically make that place sentient. I love it as a book idea, but this is real life.

What about feedback and ability to self-modify?

This would be a significant design divergence from what LLMs are, so I'd call those things something different.

But in any event that still would not actually give LLMs anything approaching: thoughts, feelings, or rationality. Or even the capability to understand what they were operating on. Again, they have none of those things and they aren't close to them. They are word completion algorithms.

Humans are not word completion algorithms. We have an internal existence and thought process that LLMs do not have and will never have.

Perhaps at some point we will have true artificial intelligence. But LLMs are not that, and they are not close.