There are plenty of executive MBA and part time MBA programs from reputable schools (not just for profit and online schools) that would allow you to work and go to school. There are some that only meet on weekends.
The one person I know that got an EMBA from a decent respected school is a manager at a FAANG. Total comp is somewhat higher than mine, but he is also on the west coast where the cost of living negates much of the difference and if I went into consultancy, it would negate all of the difference.
In many schools there is no external difference between full time and part time (evening) MBA programs. You walk out the door of both programs with the exact same degree. Nobody outside the program will ever know which path you took to get the degree.
Will they not notice the overlap between a job and the programme?
These roles are rightly challenged by many on HN since they aren't "true" technical architecture roles, for example. Even so, they do exist to support the businesses that create them until there is rationalisation during a recession when many of these roles are jettisoned.
By all general measures, company sponsorship in any significant manner is shrinking[1]. I believe the US employer education assistance tax credit is limited to around $5,250[2] per individual per year. Companies that will foot the entire bill for an executive MBA program seem to be few and far between.
[1] https://poetsandquantsforexecs.com/2014/07/22/how-to-get-an-...
[2] https://www.irs.gov/newsroom/tax-benefits-for-education-info...