> how some group would go about preserving a problem like homelessness, even unintentionally

Simplistic version: San Francisco spends roughly $100,000/year on each homeless person. In services, salaries for people working on it, rent for office buildings etc. I am willing to bet many of these people would not be homeless if we just gave them $100,000/year without all the middle bureaucracy layers.

In the homelessness example, it's not so much that the programs and groups want to justify their continued existence (though that might be happening too). It's that the programs themselves incentivize more of the problem. When they give things to homeless people, such as food, shelter, clothes, social services, even needles and a "safe place" to get high in some cases, and often with few or no conditions, they make being homeless more tolerable. Word gets around, and people who could not feasibly be homeless where they are are drawn to Portland because they will get more support there.

> would go about preserving a problem like homelessness

My state chose to outlaw homelessness [0] and to make it illegal for cities & counties to offer places to lawfully camp unless the campsites are basically enough to be KOA Campgrounds.

Actually solving homelessness is politically unacceptable, therefore it will be criminalized & preserved.

Notes:

0 - The crime is "unlawful camping".

All you have to do is promote policies that make the problem more palatable but don't actually solve it. Treatments instead of cures. As long as the problem needs treatment, the organizations dedicated to treating it will still be around, but as soon as it is no longer a problem at all, those organizations will cease to exist and everyone working on it needs a new job.

For homelessness, it's things like shelters, free meals, needle exchanges, etc. Make the life of the homeless easier without actually getting them into permanent housing. The actual solution to eliminate homelessness is to build more housing, along with financial arrangements to incentivize ownership and make it possible.

For taxes, it's accountants and tax preparation software like TurboTax or H&R block. The actual solution is to make the tax code simple enough that anyone can understand it and then withhold the appropriate amount so that nobody even needs to file. The big tax preparers have repeatedly lobbied against this.

For medicine, it's lots of prescription medication that you have to be on for life to "manage" your condition, as well as recurrent doctor's visits to diagnose and treat it. The actual solution for the vast majority of conditions that kill Americans today is diet, exercise, fiber, reduce stress, stop smoking, and wear your seatbelt. Modern medicine is also very adept at solving acute problems like broken bones, bacterial infections, appendicitis, etc. But once you've got a chronic condition of indeterminate source (many of which are actually solved by the advice above), you're in the system and can expect to pay through the nose for no solution.

For many construction projects (like say CAHSR), it's performing endless studies about how to build the project without actually building it. Some level of design is necessary for successful outcomes, but many agencies go around in circles with proposals and reports and environmental impact studies and voter referendums without actually having anyone actually pick up a shovel and start moving dirt.

You start by fixing the problem of people sleeping on benches and in tents. Then you go to those in cars, then those crashing on a couch, then those living 8 to a house, then families with small places, and so on. What the problem is keeps expanding until the resources allocated to it are spent.