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.

Three ways people respond to a problem (other than solving it)