I do think the costumes are a bit drab. I think this is because they are a bit old fashioned, nineties neutrals and cultural fusion styles.
Most modern films highlight contrasting colours for thematic reasons or to emphasise certain things, orange and teal, used to some extent with show Dany, to emphasis skin tones, explosions, night-time light and blue sky (which I hate when done with obvious colour correction but works really well when done with on-set lighting, costume and set dressing), red and green to emphasise blood, red make up, fluorescent lighting and green landscapes or just plain blue to make everything seem dreary which is used on most British films which want to appear "serious".
Then you have people who use different colours every scene to create a mood. Wes Anderson always uses a range of colour and they really create a sense of place. They may seem over the top in stills but once you get actors moving around in them they only add to the characters and setting. When Oberyn spent the whole time in bright yellow it didn't look cartoonish because after you have seen it for a second it just becomes part of his character and there is another aspect of this in that muted, pastels only suit certain people and people with a bit of melanin actually need brighter or deeper colours or they look either washed out or tired (which I do think applies to Natalie Dormer). Would a deep green dress really look strange on a noble woman in a medieval setting?
I don't understand why people think it would be cartoonish to have bright colours. Plenty of film directors use a vibrant colour palette and make it work and as there is so much bleakness against the colours it would never turn into "Attack the Block". They would also be limited to using colours with looked like they came from natural dyes so things like vivid green, purple or even true blacks for anyone but the very rich and the NW would be out anyway. One of the reasons the Lannisters and KL are so watchable is that there is sure to be plenty of interest on screen rather than just miserable grey and brown. Bright colour also works as a great contrast to emphasise the grit. Think of something like the movie Fargo. It is basically colourful all the time until something nasty happens then it switches to white/black/brown without you really noticing.
I do think that for show only, watchers when the camera goes back to the Starks or the watch, they think "Oh we are back to the boring, worthy storyline. We just have to sit through this until the fun part comes back on." There are also plenty of people who didn't know that the Freys and Greyjoys were separate families because the production team decided to go with the same colour palette for both. They wanted to do a bit of teal colour correction on the Greyjoy scenes to emphasise the sea and bleakness but at the same time cut out the gold to also add to the desolateness when that could have been used as both a contrast to add interest to the scenes and as a marker to separate them out from all the other bleakness in the show.