It's not common for researchers to find that no journal at all will accept their paper, so it's not like there's a bunch of bad science being done that we just never see.
The result of getting rid of the reputation system of prestige journals isn't obvious. It's possible that without the incentive to get into the most prestigious journal possible, many researches will lower the quality of their research. But I think it's more likely that without the hope of getting into a prestigious journal, researchers will try other tactics to coax others into reading and citing their work, and one such tactic is doing better research.
My main concern with abolishing journals is that the need for prestige in science wouldn't disappear overnight, and instead of trying to get into prestigious journals, to only way to get that prestige bonus will be to do the social climbing to associate yourself with prestigious researchers.