This question is interesting, but immensely broad.
I meant do they want the various groups, races, classes of people to get along with each other? Or do they want to divide and conquer?
Government is a tool. It's essentially a system with constraints and all participants follow the constraints and in return the government can provide "fair and equitable" treatment of all citizens. To be clear here, I quoted fair and equitable because what is fair is usually nation specific. So, for example, Canada has a nationalized healthcare system. The US does not. Americans might argue our system is unfair because the system has some structural issues and sometimes is inefficient, Canadians feels the American health system is unfair. Since very often suffering from a single illness can lead to crippling debt and bankruptcy. So, often what is "fair and equitable" is completely up the underlying society.
The only purpose of government is to facilitate the running of a society, always keeping in mind that the government will always attempt to run society in the spirit of the character of the peoples in the country.
Which one of the two approaches is more advantageous to a government and which one does the US (for example) choose to follow.
This assume that governments are monolithic, they are not. Governments are, in short, large systems with a lot of mid to small sized subsystems that may or may not be coupled to each other. The most recent election in the US was a solid example of this. The Democrats took the house, the Republicans took the senate. Then in many state legislatures flipped for one party or another. That's not to say that, systemically the government can't "divide and conquer" its own people, but it's not something that's easy to do.
What made me wonder about this is what I perceive as lack of effort on the part of the government to smooth out the relationship between the police and the civilian "class".
So up front, I know you quoted it, but I need to be explicit here. There is no civilian and police "class". There are only citizens. The police, are an extension of the legal system. The legal system is an extension of the Constitution. Which means, the job of the police is to keep order by upholding the constraints (I previously mentioned) and punishing those who break those constraints. To keep things "fair".
However, societies are complex. I won't get into the "why" I will just broadly say here, the reason the relationship between the police and some civilian communities is tenuous is, in a word: History. Simple as that. There are grievances in all directions that people feel haven't been resolved. That's normally how all these things function. It's about grievance and legitimacy.
A healthy family requires that parents treat all their children equally. Rules should apply to everyone even thou one child might be put in charge of caring for the younger siblings. If you favor one child over another, there will be resentment.
The family and children example is good and bad. It's good, because it should represent the best humanity as whole should aim for. However, in political terms, families are authoritarian, benevolent, but authoritarian. The analogy is mapping to someone is the "parents" and thus "knows what's best" for the kids. Which, from the perspective of a citizen, might be alarming. Imagine the government coming in and tell you who to marry, what job to take, what to learn at school. That's parenting. Making life changing decisions for people for whom you think have no idea what they want. Children, while it may be true that they don't always know what they want, still have agency. They can grow to resent parents who practice strict discipline and sometimes, they'll ignore their parents because they feel their parents have no clue about what they want.
Which bring me full circle to legitimacy. Governments are about legitimacy. What governments need from citizens is participation. What citizens need from government is to act in legitimate terms. So if a person / party decides to destroy norms or break the rules to get things done, in the short term that works. But, it destroys the legitimacy of the government and then over time, it erodes the faith people have in institutions. Which eventually leads to collapse or a failed state. Legitimacy is everything and it's very important. If a party undermines democracy, or worse, democracy no longer reflects the will of the majority, it slowly destroys the legitimacy of the state.
So, what are governments trying to do today? In my opinion, they're trying to prevent the bleed of legitimacy. Because increased polarization is slowly destroying democracy and as people become more polarized they more and more see governments as illegitimate. So, this is why centrism and compromise are important. Because it appeals to the most citizens, which gives the government legitimacy. If a government just rams through policy for a particular party, then they'll lose legitimacy and eventually their country. Simple as that. Legitimacy is about trust, which is hard to come by, and VERY easy to lose.
TLDR: Broad answer for broad question.