I think that it is seen as extreme by a number but remember this is also an internet petition which limits the population that can participate (not everyone is internet savie enough). My father is a RSPB member follows birdguides etc. but had not heard of the petition. I found it through here and would otherwise have been oblivious.
Anti-bloodsports would be behind it 100% but birders are a more conservative, apathetic, crowd so by the nature of the petition, being extreme, shows there is quite a lot of feeling out there.
A number of RSPB membership that you are counting aren't avid birdwatchers - more casual and some overlap aswell with the hunting fraternity.
Anyway, it would be interesting how many the 'support' grouse shooting petition gets - I haven't looked.
Not everyone is on the Internet as you say
If a couple share an e-mail address only one can sign as it then says to the 2nd you have signed
If not on social media people are only just becoming aware depending if they watch the right news channel or paper
Joan