It may be the culture in some ecosystems but it's entirely false. A nonprofit can pay competitively with organizations of its size. I've worked at nonprofits and I've seen this line towed far too many times as an excuse to lower labor costs for people producing value for the organization. If your nonprofit is service oriented in anyway and requires people, appropriately investing in those people through their compensation and WLB is an appropriate investment.
In theory, a nonprofit should be able to outperform and outpay a for-profit entity of the same size, largely because there shouldn't be an expectation of the organization to cut a big slice out to investors but instead to invest back into itself. This could mean increased hiring, grabbing top talent, investing back in a cause and so on.
Very often, the level of funding is what it is and the non-profit has very little control over it. If you choose to pay higher salaries, you get less work done. This is common in the academia, which is usually funded by taxes and tuition fees.
Sometimes there is even an inverse correlation between funding and salaries. If a charity chooses to pay higher salaries, its "administrative costs" increase. Donations may then dry up, because people consider the charity inefficient.
Many non-profits rely extensively on volunteer labor. Salaried professionals often contribute their time and expertise to a worthy cause for free. That puts paid employees in an awkward position. It's hard to argue that you deserve a higher salary when the market rate for your services is 0.
That feels like how it works outside of academia as well.
In the industry, your employer tends to own the projects you work on.
First, if you choose to work for a non-profit, you should expect a huge pay cut. That's the way the society works. Or actually it's the opposite. If you work in a for-profit role, the society allows your employer to keep most of the value you create. If you create a lot of value, that gives you the leverage to negotiate much higher compensation than your peers in less profitable roles.
Second, like in any business, you should focus on what brings the money in if you want to advance your career. You should be the PI instead of the lab technician. Instead of building the software other people ask, you should become an expert in the field and build the software other people will need in the future. For example, bioinformatics as a whole used to be more of a support role, but today there are many high-profile bioinformatics PIs.
Third, many people in the industry fail to realize that you don't really work for your employer in the academia. You work to build a reputation for yourself, which makes you valuable to the employer. If you want to continue working for other people, the upward career path from support roles goes to the administration.