You're assigning agency to a notation, which is not very productive. Big-O doesn't "assume" anything, doesn't depend on what "matters", or try to achieve anything. It's well defined and you need to understand what Big-O is to understand its result.

If you want a better overview of algorithms for practical reasons, you can use o(), O(), average case and ammortised complexity, or just provide histograms of runtimes for algorithms you're comparing.