I agree the fact they were all asked for approval makes little sense, even Davos points it out. But there was some logic to why they were there. Technically Jon was the Lord of the North anyway so makes sense his family would come down to support.
That would explain their presence in the City (though I guess they kind of forgot about the good old "there must always be a Stark in Winterfell" mantra), it doesn't explain why they all get to sit on the council. When for example the notoriously rebellious Dorne accepts to be reduced to one seat, accepts the North's independence, without demanding anything for themselves and accepts a crippled Stark boy with the personality of a slice of Toast as their ruler.