Solving them at scale can still lead to money. Sometimes they are willing to give you a little money, sometimes others are (governments, or even businesses if it can lead to profit for them. I.e., low income training + job search with recruiter fee could be profitable). Or if you're addressing the problem of they have little money, by giving them skills or opportunities, you can charge after you help them (i.e., many coding bootcamps that delay costs until you land something)
Payday loan revenue in the rich burbs roughly-zero, Payday loan revenue nation wide mysteriously between $6B and $46B depending on data source
Now can you get a cool and trendy very high social status hipster job to brag about on social media in the coolest office with only the best "personality fit" coworkers helping poor people, oh heck no you'd be savaged here if you admitted to working for McD, but the market of poor people products, at least in some areas of business, is about ten times as large as rich people products.
There are a hell of a lot more poor people than there are rich people.
It’s also a hell of a lot easier to take advantage of them, given they generally lack the education or resources to protect themselves from predatory businesses. I struggle to think of McDonald’s as “helping” poor people. It’s putting food in their mouths, yes, but if McDonald’s were more concerned with the health and financial wellbeing of their customers as opposed to exploiting psychology to get as many people to impulse buy as much junk food as possible it would be an incredibly different company.
The goal of a company is not “help” its customers, the goal of a company is to exploit any and every untapped source of wealth. If it can do that by hurting someone who doesn’t know better, it is going to.
Your first statement seems to imply that the market is flawed (because, obviously, the willingness to pay for something is going to be higher for someone with lots of money than someone with little), the second seems to imply that there is something inherently beneficial in focusing on 'market solutions' as you define them. These views are rather at odds with each other.
But, I'm not trying to comment on either of those, just that you can create utility for a poor person and still have a viable business. If it relies on government for funding, because society has decided it's worth funding that sort of work, so what? We have an entire defense industry in the US based on that idea. Most of our education is too. Other countries have their entire healthcare systems based on that idea. You can decide you don't want to be paid directly by taxed money, but that's on you, not on the viability of such a model.
To be clear, I’m criticizing the common notion that the market is inherently beneficial or that it contains the solutions to all problems (though I don’t mean to accuse you of holding such views). Some problems necessarily require non-market solutions.