Air Iceland do indeed fly to Kulusuk (KUS) and Neerlerit Inaat (CNP) all year, and also to Kangerlussuaq (SFJ) and Narsarsuaq (UAK) in the Summer. The flights are irregular (sometimes once weekly), and frequently interrupted by weather.

Frankly, if you're making the effort to go to Greenland, it's a waste of time to stay on the East Coast, though (KUS and CNP). Ilulissat Kangerlua is a must for all visits to the country!

Air Greenland do fly KUS-SFJ. Domestic airfares are expensive, though (and it's that or sled in the winter months - boat is not an option).

Your only other option (other than going via CPH) is to look for First Air charters from Iqaluit (YFB) to Nuuk (GOH) and Aasiaat (JEG) in the Summer.