I think you do not understand what problems that might cause once the user does this and does not understand they cannot go back and then they reopen the file and start looking for the number they had for the artboard with a certain art on it an they now can no longer find it, they are going to start screaming bloody murder.
This has to be thought out better then just renumber the art boards but reordering the pages numbers automatically.
The users has to make a definitive decision to do so.
Just think about it?
understand you think well it's a no brainer but that is not the case this matter has been discussed and there are problems with this scenario you are not thinking about.
The whole point of being able to move the artboards with the content is so you have maximum flexibility, once you have the automatic renumbering you loose that ability even though you can sort of physically move the boards because you loose the number of the board.
This has to be done by a command to renumber the artboards and a warning are you sure that you want to do this and that if you do you loose all previous numbers and once save it cannot be undone.
I say the ability to renumber and rename a board is a good one because that is both artboard and art specific, you know which board you have renumbered!
I am against a auto renumbering system, even as an option it is a dangerous move.