Cookson had been quoted by Cycle Sport magazine as saying that Team Sky "isn't quite staying true to one of the reasons it was formed".
However, Drake has insisted that the British core of the team remains intact.
"In my view, it's very much a British team. Generally, around a third of the riders each year have been British. The most it has ever had is 10 and now there are eight," Drake told Cycling Weekly.
"If you go back to why we felt we needed a pro team in the first place that linked to the national federation, it's because we were concerned that we had nurtured all these riders since they were teenagers and then we told them to go into a murky world.
"I think Team Sky has been instrumental in creating a safer environment across teams, and with the talent path we have in place, we are never going to have all those riders in the same team. There are more British riders with more opportunities, which shows you the state in which the sport is in."