It is probably “incorrect”, but given the established actual usage over the decades, it’s most likely what you need to do nevertheless.
Not doing it is like punishing people for not using Oxford commas, or entering an hour long debate each time someone writes “would of” instead of “would have”. It grinds my gears too, but I have different hills to die on.
Not sure I agree. The correct thing is to not mess with the URL at all if you're unsure about what to be doing to it. Doing nothing is the easiest thing of them all, why not do that?
If different clients does it differently, you have incompatibilies. This punishes everybody. Since normalizing // to / removes information which may be significant, the obviously correct choice is folllowing the spec.