With sufficient randomness, is XOR an acceptable mechanism for encrypting? I have heard criticism of various cryptosystems saying that "at their heart, they were just XOR."

Is this just ignorance, or is there something inherently wrong with XOR based ciphers?
It's important to make the distinction between ciphers which use XOR
internally as a component operation (which is nearly all of them),
and 'ciphers' which just XOR the plaintext with a secret. If the key
is the same length as the plaintext, then it's a one time pad, so in
some sense, yes, with "sufficient randomness" you can safely encrypt with XOR.
The problem with this is that it only applies if the key is
exactly as long as the plaintext, and is perfectly uniform random.
If it's not, then you've basically created a Vigenere
cipher, which
can be trivially broken. It's interesting, and unfortunate, how
the 'provable security' of the OTP fools people into thinking
they can just take a little shortcut or two and still get
something safe.

Typically, people who implement simple XOR ciphers compound their
mistake by using hard-coded keys, in which case a single known
plaintext/ciphertext pair is sufficient to decrypt anything.
That's probably mostly because if you were some kind of prebuild
crypto toolkit, it would be easier to just use AES along with your
RSA key management, etc, versus using a decent key distribution
scheme and then using XOR with your safely shared keys.