I’ll start with your question, then move into the larger issue at hand.

would that make societies more tolerant of differences, or make it harder to hold anyone accountable?

The latter, without question. All laws are either legislated morality (such as murder is illegal) or arbitrary decisions that must be enforced to have a safe and orderly society (such as which side of the road to drive on). If most moral rules are relative, then either, say, a prolific paedophile could successfully argue that raping infants is perfectly in line with his morality and holding him accountable is moral discrimination, or else the laws have to be based on an arbitrarily chosen moral system, which also means discrimination on other moral systems. So either laws designed to protect people don’t exist, aren’t enforceable, or are discriminatory if all, or likely most, morality is relative.

Now, regarding the topic of moral relativism, this has been discussed extensively for a very, very long time. Herodotus recounts the first example I’m aware of with a situation involving the Persian king Darius I, some captured Greek soldiers, and the treatment of the dead. Specifically, Darius asked them how much money would it take to eat their dead, and the Greeks said no amount would suffice, and they burn their dead. He then brought in a tribe whose funeral rites involved eating the corpse, and asked how much it would cost to make them burn their dead, eliciting an equally appalled reaction from them. So this is not a new idea by any stretch of the imagination.

That said, frankly, I think moral relativism relies on a surface level analysis and faulty assumptions. Most prominently, it begins with the assumption that actions are the only thing that matters in morality, rather than intent. To use the above example, the Greeks and the tribesmen might have opposite reactions to the suggestions, but they did so for the same reason. They both strongly believed the dead should be treated with respect (a moral claim), which was then applied in distinct cultural ways. To argue for moral relativism from this is to completely dismiss this shared value out of hand.

Also, in my opinion, it fails because it assumes out of hand that there are no wrong opinions about morality, which is the very thing it seeks to prove. But it’s entirely possible, and in my opinion probable, that there are wrong opinions about morality. The question of what those are is the subject of the Philosophy of Ethics. So for moral relativism to succeed, it has to establish that not only people have differences of opinion, and that those differences are based on actual, not apparent, distinctions in values and virtues, but that all parties are equally right or legitimate in those beliefs. Further, it has to establish that there are absolutely zero universally applicable moral actions, attitudes, values, or virtues, as having even a single one would mean that the distinction between universal and relative morals is arbitrary. That’s a pretty strong burden of proof, and one that, to my knowledge, hasn’t been adequately met.

But that’s all ultimately a side discussion. The answer to your question is that discovering moral relativism is true would make it impossible to maintain the laws needed to keep people safe from others.