Wielding a staff-group weapon, that isn't a quarterstaff, (it is in fact part of a quarterstaff) in each hand.
If staff-group weapons are all staff-implements, then it works.
But staff-implements state that they are quarterstaffs. So you can't have a staff-implement that is "the offhand end of a quarterstaff"
Alright.
"staff-implement" as you mean it is an item available in the PHB's equipment section for 5 gold pieces. It has the ability to be used as a staff implement, and also qualifies as a quarterstaff.
Quarterstaffs are members of the staff group of weapons.
What you are saying then is that somehow a feat could turn a five-gold piece stick into a double ended quarterstaff, and because each end has the weapon type staff, that therefore they are that five-gold piece stick... logically extended that must mean that you have an infinite recursion and an infinite number of staffs and five gold sticks...
Sorry. Your misunderstanding of what's going on here has caused you to go into a rediculous direction, and use that rediculous direction as the source of an argument against something.
The problem here, is that you think that 'Anything that can be used as a staff' must therefore be a five gold stick available in the Player's Handbook. The very existance of superior implements says, no, that is not the case. A 30 gold stick is not also a 5 gold stick.
What IS happening is:
You have a 5 gold stick called a quarterstaff that is of the staff weapon group. You have a 5 gold stick called a staff that can be used as a quarter staff.
A member of the staff weapon group is not the same thing as that 5 gold stick in exactly the same way that a weapon of the mace group is not the same thing as the weapon 'mace'.
If you can understand 'mace' you can understand 'staff'. It isn't hard.
The thing is, class features don't mention the five gold stick. They simply mention 'staff.' As staff is a weapon group, the same as 'light blade' and 'heavy blade' and no text exists to the contrary, you therefore have no exceptions to the general rules 'Quarterstaff is a staff' and 'wizards can use staffs.'
Seriously. If there IS no exception, APPLY THE DAMN RULE.