@Michael
<< If you use Edit> Separate on the polyline it should then work like you're asking I think.
Yes fine!

If i good remember when 2 points or more are overlaped in the same x,y,z
there was a red square... but don't remember in what Function(s) and if it's always living! :)

@ Tom
Seems you are confused by "Control Points" and Points who are possibly on a curve!