Pity is compassion for the suffering of others.
Empathy is understanding the feelings of others.
Both of those things are different forms of empathy.
Empathy is a set of related social skills encompassing not just the ability to understand another person's perspective but also to respond with the appropriate emotion in order to try and meet that person's needs.
Like, lets consider the plight of the homeless. People might say we should have empathy for the homeless and help them. But a high percentage of the homeless have substance abuse problems, who don't want to be in shelters because they can't do drugs there, their addiction drives them away from help. Empathizing with that person isn't a bad thing, but it doesn't lead you to getting them off the streets and free of drugs.
Empathetic concern (or pity) is not voluntarily. It's not something you can turn on or off, even if you want to. You either have it or you don't, and if you don't have it it's because you never developed the requisite skills.
It is entirely possible to feel pity for someone and simultaneously to understand that it is not within your power to help them.
The reason many people who are sleeping rough have problems with substance abuse is not because they are bad people born without souls, it's because they have deeper problems which they are self-medicating. Many will be victims of various forms of abuse, either before or after they became homeless. Many will have undiagnosed and untreated mental health issues. All of them will be accustomed to poverty and hardship. There are typically a long series of societal failures which result in a person ending up in that position.
And this might lead us to ask other questions. Why do we expect people living in horrendous conditions to become abstinent on their own before we can help them? Why not allow people to do drugs in shelters under supervision? Why focus on abstinance as the only acceptable outcome rather than focusing on mitigating the harm addicts cause to themselves and others?
The problems you have described are very difficult to fix, but empathy, understanding why people are the way they are and being willing to consider and attempt to meet their needs rather than the other way around, is a necessary first step.
You don't see the people preaching empathy showing empathy for conservatives or Christians or the wealthy.
By your own definition of empathy, you do. What you don't see is empathetic concern, because empathetic concern is kind of specific to people who are suffering.
Again, empathy is involuntary and thus can be complex. You can empathize with someone, or even feel empathetic concern towards them, and yet recognize that they are harming other people.
But here we have a situation where a mother finds out her son's middle school classroom has "This Book is Gay" available for students, a book written by an adult provided by a school run by adults which contains literal instructions on how to give better handjobs, and people have absolutely no empathy at all for that woman or her concerns.
Again, you can empathize with someone and yet recognize that they are harming other people.
Any real understanding of that woman's perspective will come to the conclusion that it is based on bigotry towards others and a desire for control over her son's sexuality which is ultimately harmful to everyone involved.
Non-coincidentally, a very high proportion of homeless people are also gay, and end up in that position due to parental rejection.