In most STEM fields I think it works better the other way around: an undergraduate degree in the STEM discipline and then an MBA. Or (if feasible) both an M.S. in the STEM field and an MBA.
My thought process was when young and starting out in work life (after short military service out of HS) a business degree had more applicability across multiple fields/sectors and wasn’t necessarily going to put me only in a CS job as the first CS degree most likely would. Also, with the way tech changes, I wanted the CS degree to be as near current tech as was available when I got it. My first boss (or dept mgr who was my boss’s boss) had a MS in CS from what was then nearly 25 years in the past at that point and hadn’t really kept up with tech and most of his knowledge was old and it was sort of obvious he wasn’t really keeping up with his dept and just marking time until he could retire. But all that said, if I did it over again and suspected my entire career would be in some flavor of IT which it has been, then you are right and a CS in both undergrad and grad might be best while also adding mba and pm certs much later.
Similar to what I did - I have a major in Physica and a minor in business. Thankfully it worked out for me, over the years I’ve gotten jobs due to the value that someone with a STEM degree brings - Problem solving, etc. I also learned the folks that understand that connection usually end up being good bosses! Business side was useful once I was already in the position and helped me move up. (Currently in a procurement-related field and loving it)
My 25 year old son has a BS in Biotechnology. His employer is paying for Masters in Engineering Management from UMass. He works for a major pharma corp in MA. They would not pay for an MBA.
Most of the jobs listed are not entry level jobs. You do not become “Sales Manager” unless you were a sales person to start. In most cases the Peter Principle was applied. At lest that has been my experience in my industry.