I'd argue it actually makes things worse. When you can have a higher-purpose job (an ICU or ER nurse who is saving patient's life everyday) and you're spending most of energy on administrative overhead, the effect is just magnified.
Meaning is a subjective thing. That's why some people thrive in some environments and some may burn out. If you put your average IRS auditor in a hospital, they might actually find more meaning in filling forms than exchanging with patients.
That'd mean that people who are burned out all did so because they did stuff that didn't have meaning? Ultimately, I think you can get burned out regardless of how meaningful it is or isn't. People working at hospitals (just as one example) have probably some of the most meaningful jobs, yet burn out frequently regardless.
More likely that both different people burn out because of different things, and it's a mix of reasons, not just one "core" reason we can point at and say "That's why burnout happen, not the other things".