Hell no. Been supporting City since the 80s, I became a fan due to the warmth and togetherness, win or lose. Felt like a family. My actual family were partly raggies and desperate for me to go along to Stretford with them, just couldn't understand it when I refused.
We were often called 'sleeping giants' back then, so our support and reputation were recognised anyway, but what would being a 'big club' even mean nowadays? The best social media presence? The most online fans? Most likes on a post? Most trophies won in a specific timeframe? Most nationalities welcomed onto the stadium tour?
City is City. It's my City, and whether person x y or z likes that or not, I don't care. Just like fans of Cheltenham, Ipswich, Carlisle or any club shouldn't care.
Besides, a major flaw of thinking you're a big club is that you start to belittle other clubs just for existing. One of the main reasons fans of almost every other club dislike the scum, thanks to their attitude in the 90s and beyond. Let's not be them.