People love to point out utilitarianism is bad by using reductio ad absurdum. However, the point to utilitarianism to me is not to have some kind of mathematical function to calculate utility, and to follow it slavishly. Rather, it is to be pragmatic and to
not be anti-utilitarian.
There are many cases where the "ethical" course of action makes many people miserable and where a simple, pragmatic, utilitarian solution would be beneficial. You don't have to go to extreme examples like the trolley problem or shooting down planes with terrorists in them. Just look at organ donations, for example. Here in Germany, it is opt-in. Why can't we just make it opt-out, or like in the US: when you get a drivers license, you have to decide yes or no?
I would consider myself somwhat utilitarian, but I have no problem reconciling most of the mentioned problems with common sense. Sometimes, it is by questioning the set-up:
> b. Framing the innocent
> You’re the sheriff in a town where people are upset about a recent crime. If no one is punished, there will be riots. You can’t find the real criminal. Should you frame an innocent person, causing him to be unjustly punished, thus preventing the greater harm that would be caused by the riots?
The real problem here is that people are going to riot if their sense of justice is not satisfied. You're not just a sheriff but also a philosopher - the solution is to change society!
Most of the problems are variations of the classic "one person suffers greatly so many can have shallow pleasure". I don't know anybody who really proposes integrating over utility without any weight like this. Instead, most people would weight the injustice much greater than the pleasure. I maintain for example, we could radically reduce CO2 emissions and redistribute wealth to reduce poverty if we were willing to accept a couple percent GDP degrowth. That means on average, you might have to wait 2-3 years to get a new phone or a new car, and you might have to take a less than ideal job for a couple of years. But your inconvenience would be more than compensated by the increased utility for all.