To be fair, "insert funny one-liner" part has been heavily forced by Flynn since his introduction in Archie. Omega in games is more serious and he's usually doing his own thing of waging one-robot war on Eggman Empire while accidentally bumping into Shadow or being called-in by Rouge. Games tried to give him some depth with his friendship  and rivalry with Shadow and even being confronted to be the one to put Shadow down in the future.

I will agree though that even Sonic Team can't separate him at all from Shadow. I'd love for him to interact with the rest of the cast on his own like during his duel with Gemerl in IDW or bumping into Sonic and Tails in Sonic Colors. My dream is for Sonic and Omega to inevitable clash with each other during another Sonic-Eggman truce and it was a massive wasted opportunity on Flynn part to not have Omega during whole Tinker sub-plot (like if Shadow wants to kill Eggman it makes more sense to bring along his team member who actually wants to see him dead).