Smaller shaft isn't going to help with spin. If you want to get more spin with the same cue position, put a smaller radius on the tip. It will make it more sensitive AND keep the hit closer to the center line of the cue.
A bigger shaft has disadvantages shooting a frozen ball off a rail and when putting draw on the ball, simply because to hit the contact point you want on the cue ball you start running into the rail or the slate. With a 13mm+ shaft, you need to get quite close to the table and/or angle the cue down to get a powerful draw, and that can be awkward on the bridge hand. A smaller tip radius actually hurts you a bit with the ball frozen to a rail unless you're looking for lots of follow.
It's all a balancing act. I like a relatively stiff taper and something under 12.75mm for the tip....12.6, maybe...something like that, and always with a dime radius. The tiny little tips you see are usually on low deflection cues, and they're like that just to make the tip end as light as possible. It's nothing to do with spin.
Best I can tell, 12.75 is the new standard. When I was growing up, it was always 13mm. Lots of pros used to play with a 13mm, or even bigger. I think Ray Martin used to play with a 13.25mm. That's the same as my break cue, which feels absolutely ENORMOUS to me.