Yes, we're doomed. In fact, I don't think the types of errors you are pointing out aren't even worth worrying about, given that even the BBC can't seem to reliably choice between "to" and "too". "There's" (instead of "there are") seems to be used and accepted by more or less everybody these days. And don't get me started on its/it's, there/they're, etc.
Yes, we're doomed. Luckily, in almost all cases, it doesn't really matter.