Do we need symmetric cryptosystems?
Yes, we need symmetric cryptosystems, for many reasons; to give three of these:


We need a hash function to make most asymmetric cryptosystems secure (e.g. we simply do not have a secure signature system based on RSA without a hash), and current hash functions are (or are built from) symmetric cryptosystems.
All asymmetric encryption cryptosystems are bound to output ciphertext larger than the plaintext, and there are situations where this is a major drawback (e.g. bulk disk encryption).
The best asymmetric encryption cryptosystems we have around are some orders of magnitude slower/more energy-hungry than state-of-the-art symmetric ones; it turns out this worsens, badly, if we want to minimize the ciphertext expansion ratio by using huge parameters.


Note: Point 1 has the appearance of being sufficient to prove my assertion, but it is debatable to a degree: perhaps in this question we want the big picture only, and disregard the symmetric internals of all existing asymmetric cryptosystems; and/or perhaps we could devise asymmetric-based hashes (though these would be symmetric cryptosystems in disguise).
Also: we could live with 2, if there was not 3, which in the end is the real killer.