Earlier quoted context omitted.
MS-DOS is maybe the ugliest operating system I ever had the displeasure of using. There's very little to learn from operating system design point of view in there, and better sources are available under more permissive licenses -- ie open source licenses. It looks like a PR stunt to me.
'Operating system' is stretching it. 'Disk Operating System' is one of those phrases like 'ice creme' that implies 'ice cream'. MS-DOS was a program that wrote stuff to disk for you, read the keyboard for you, listened to interrupts for things and showed useful error messages such as 'Retry, Fail or Abort?'. There was no networking to speak of, certainly not TCP/IP as we know it, no user permission things, you couldn…
The BBC micro's operating system had no networking to speak of, certainly not TCP/IP as we know it, no user permission things (indeed, no concept of users at all), and no cron jobs (indeed, no OS scheduling of tasks at all). In other words, not one of the things you specifically called out as "normal stuff that an Operating System does".
(There was a networking system, called Econet, but it wasn't part of the base OS; you needed an extra ROM -- physically plugged into the circuit board, those were the days -- that implemented it.)
For the avoidance of doubt: I loved the Acorn MOS and hated MS-DOS. But it simply isn't true that the BBC Micro had those features and MS-DOS didn't. And, for what it's worth, I see nothing wrong with calling either of them an operating system.