There is a major gap in this analysis by not controlling for industry or companies. Engineering Manager is a very generic title, so this is going to get Start Ups, Big Tech, Little Tech, Enterprise, Contract Shops, etc. Staff Engineer is very uncommon in Enterprise or Contract shops, there you typically see SWE 1/2/3 -> Tech Lead -> Architect. Most Tech companies I think have more of a SWE 1/2/3 -> Staff Engineer -> Principal.
The other part is that Engineering Manager is a terminal position, I've known a few people who were manager for 20 years without ever going to Director / Exec whatever, its just a competitive jump and mathematically most will never go up. This is ALSO true for Senior -> Staff and Principle though. But Engineering Manager positions often have more of an upside with bonuses / incentives than Engineers get.
Finally it is ultimately a career change, and that should be the primary factor to consider.
The other part is that Engineering Manager is a terminal position, I've known a few people who were manager for 20 years without ever going to Director / Exec whatever, its just a competitive jump and mathematically most will never go up. This is ALSO true for Senior -> Staff and Principle though. But Engineering Manager positions often have more of an upside with bonuses / incentives than Engineers get.
Finally it is ultimately a career change, and that should be the primary factor to consider.