They think of themselves as football royalty.
In that article sent to Jacob Reese Mogg it said that 'traditionally' their fans would sing 'god save our team' when singing the national anthem - they really do think that they are royalty.
I'm not sure what the measure is of a 'big club' - but it seems to be defined by silverware above all other things (stadium capacity, following, expensive players, history etc etc).... The term seems to have redefined itself.
I remember when Forest were a big club, the biggest club in Europe for a while. And, as for history, isn't Nottingham County the oldest club in the league?