You find a block, what do you do? A rational miner will take all transaction with fee >0 (greater than zero but not zero), even if only 1 satoshi. Otherwise the miner who finds the next block will take it. This is how competition works, and this is one of the best features of bitcoin.
That is not correct. While competition will keep fees down miners do face orphan cost. Larger blocks means longer propagation time which means greater % of blocks lost to orphans. It doesn't make economic sense to add transactions who's fee is below the orphan cost. How large the costs are and how rational miners act can be debated but a fee of 1 satoshi is for all intents and purposes a fee of nothing.