Yes, but horses and cows also compete for the same building blocks. And yet, we have horses and cows.
Your argument could be right if there wasn't an abundance of these building blocks.
I think you're just failing to grasp the sheer numbers involved here: there are probably around 10^9 cow-like critters and they interact with other, competing critters every, let's say, 10^4 seconds, with the odds of any one of these interactions being fatal being, again, let's say 1 in 10^4. By comparison, there were probably something like 10^37 (if I counted my orders of magnitude right) biomolecules interacting every 10^-6 seconds where every time some new chemical "A-ha!" moment occurred, that reaction now became 1000x more favorable than any other competing reaction. That's like, what, 41 orders of magnitude; even if I'm off by 10+ orders of magnitude, there was just a whole lot more selective pressure going on during the pre-biotic era. Less favorable chemical pathways were outcompeted on scales so vast you'd need trillions of cow-years to approximate how de-de-de-deselected-for they were.