The way I understand it, since the data that's being hashed is pretty much random and because the hashing algorithm exhibits the 'avalanche effect' it probably doesn't matter if you keep starting with 1 and incrementing it or if you use pseudo random values instead, but I was wondering if anyone could support this or disprove it.
Yep, your understanding here is correct. It does not matter what exactly gets hashed, and no, you can't cheat without first breaking SHA-256, which is considered difficult.
The salient property of cryptographic hash functions is that they are as random as is possible while still being deterministic. That's what their strength depends on -- after all if they weren't random, if there were obvious patterns, they could be broken that way. So the ideal hash function behaves just like a random number generator. It does not matter what you feed in, timestamp or not, whatever's put in there, the hash should still behave randomly (i.e. every possible outcome has the same a-priori probability of occuring). Incrementing by one works just as well as completely changing everything every step (this follows from the avalanche property). However, the initial value, before you start incrementing, must be (pseudo-)randomly chosen, or every computer will start at the same point, and the fastest one always wins, which is not what is wanted here.