When I got my Bandit it had a little less than 2,800 miles on it. I ASSUME it had a standard dino oil in it when I got it ( It was nice and clean ). The 1st change I did was a " Blend " and every change after was a full synthetic. I felt that the shifting of the Bandit was great from day one so any "perceived " increase in smoothness could be contributed to an issue of perception.
I've never used manufacturer labeled oil or Dollar Store oil. Even in the old day's I used standard high performance car oils. A lot of the time it was Valvoline 20w 50, same stuff I'd put in my sports cars.
My point is, don't over think it. Any quality oil that meets or exceeds the manufacturers recommendation, be it dino or synthetic, will serve you well. Buy a quality oil, change it and your filters regularly and you'll be fine.
BTW - I use Mobil 1 or Castrol. I have to buy separate oil for the bikes since all my other vehicles run 5w 30.