If a non-quarterstaff staff weapon group item is an implement, it cannot be a staff implement because a staff implement cannot be a non-quarterstaff.
This is based on an incorrect premise.
The following is true:
All items marked 'Arcane Implement' in staff form are quarterstaffs.
All items enchanted with 'Staff Implement' enchantments are quarterstaffs.
No where does it say that all items that can be used as staff implements are quarterstaffs.
Here's the argument defeated by reductio ad absurdum:
All items marked 'Staff implement' cost 5 gold.
All quickbeam staffs cost 30 gold.
Therefore, a quickbeam staff is not a staff implement.
Notice... the argument uses the same form, using a single member of the staff set and claiming that its properties apply to ALL members of the staff set.
Also, the argument is defeated another way:
Your logic is:
A quarterstaff is a staff
Staff fighting makes it a double weapon. Each end is also a staff.
Each staff must therefore be a quarterstaff.
Therefore it does not work.
ALL you have done is proven a quarterstaff is, in fact, a quarterstaff and that each end of the quarterstaff can be used as a quarterstaff.
To which I say 'Well, thanks for proving a tautology.'
Moreover, it is possible to have a staff implement that is not capable of being used as a quarterstaff:
Corellon’s Implement (11th level): Choose an arcane implement that you specialize in, whether the wand, staff, or orb. You can use a longsword as if it were that type of arcane implement when casting your spells.
In this case, the longsword does not benefit from staff fighting, does not become a +2 proficiency bonus. However, it is still, should you choose it, a staff implement. Nor does it contradict the rule saying that the 5 gold implement you buy in the PHB equipment guide can be used as an implement.
Therefore, it does act as proof that 'all staff implements must be entire quarterstaffs' is utterly incorrect which means that yes, you -can- have two ends of a weapon that are both staffs and not have both be quarterstaffs, and have them both be useable as seperate implements.
tl;dr: The text refering to 'staff implements being quarterstaffs' refers to a -very specific object- that is available on a very specific table that costs 5 gold. You cannot take the text that refers to that specific object and claim with any rationality that it applies to all objects ever.