Having read the basics of mining and having also read SHA-256: (Probabilistic?) partial preimage possible? and http://jheusser.github.io/2013/02/03/satcoin.html

  I have the following question:

wouldn't it be better to select a random nonce instead of increment it by one? This way, the miner would make use of the birthday attack. Which would make the probabilities of finding it higher (for a given number of trials)

I'm not sure my understanding is correct, so maybe my question sounds stupid.

Having read the basics of mining and this about SHA256 (probabilistic?) partial preimage possible? and SAT solving - An alternative to brute force bitcoin mining, I have the following question:

Wouldn't it be better to select a random nonce instead of incrementing the first by one? This way, the miner would make use of the birthday attack, which would make the probabilities of finding it higher (for a given number of trials).

I'm not sure that my understanding is correct.