I don't think you can easily have both, and this problem underlies much contemporary political conflict. At its core is a pervasive hypocrisy: people readily engage in broad, often negative, stereotyping about groups they oppose, or are not a member of, while aggressively resisting and resenting any attempt to stereotype about their own group, or groups they favour.

So, in the aftermath of a terrorist attack, some people will make broad statements about Muslims, and the need for collective action within Islam to address religiously motivated violence. Others will reject this, and say Muslims in general cannot be held responsible for the behaviour of a minority of extremists.

Then, in the aftermath of a sexual harassment scandal, some people will make broad statements about men, and the need for collective action by men as a gender to address sexual harassment and sexual violence. Others will reject this, and say men in general cannot be held responsible for the behaviour of a minority of harassers.

Oftentimes, the same people will be on the complete opposite sides of each debate. And they will unthinkingly employ language and rhetorical tactics in one that they would utterly reject in the other. This behavior can be seen in almost all discussions across or about political affiliation, race, religion, gender, sexuality, and just about any other cleavage you can think of.

This is the real empathy gap. People reflexively respond to any criticism of their own groups with hostility, while attacking others without restraint, and dismissing hostility from members of that group as an overreaction. They recognise a degree of individuality and diversity within their own favored groups that they simply can't or won't recognise in others. They use tactics against their enemies that infuriate them when used against themselves. In some cases, they may recognise that those tactics are wrong in specific cases, and even in the abstract, but defend their use by themselves or their allies with excuses based on cambatting political correctness or appeals to historical victimhood.

Making all of this worse is that statistical thinking about groups is a necessary of component of any mature political debate. Certain traits or behaviors often are more prevalent within some groups, for a variety of reasons, and being able to recognise and talk about this is basic prerequisite for any attempt to address social and political problems.

In theory, people should have no problem recognising statistical differences in a group, while also recognising and respecting the inherent right and diversity of individuals within that group. In theory, people should be able to hear comment, even criticism, about their own groups, and judge it purely on its statistical reality, without feeling that it is a personal slight upon them. But in practice, this doesn't happen.

Besides the reflexivite hostility discussed, statistical thinking often does give way to blind stereotyping amongst outside observers, and this stereotyping gives rise to discrimination. Causal factors are ignored, or distorted, and adaptive traits are portrayed and inherent, and vice versa. Instead of helping to inform rational debate, this discrimination entrenches and exacerbates division. Soon, any statistical thinking about a group that carries an implied or perceived criticism becomes tantamount to discrimination, and groups become hypersensitive to it.

All of this is probably just an inherent flaw in human cognition, and likely unfixable, without some kind of attempt at species-wide genetic engineering that completely reconfigures our co-operative behavior.