"designed to function correctly with our genuine inks"
That comment is arse about face: it is the inks which are designed to work with the printers, not the other way round.
If some third party makes an ink that confirms to the printer spec - then it can be used. The printer should not care.
Imagine the brouhaha if Ford cars only worked with Ford petrol, Ford oils, Ford tyres, ...
(Other car makes are available)