Turning PRPs in PRGs with a counter
You are quite correct.  A PRP in counter mode is, in fact, distinguishable from a random sequence if you approach the "birthday bound".  We get around this by never generating that much output at once.  With a 128 bit block cipher, an output of $2^{40}$ bytes (which is a lot of output) gives us a distinguishing advantage of about $2^{-56}$ (the probability of a random sequence of blocks of that length would have a repeat) -- this is small enough that we don't worry about it.