I am on the fence about this. The last thing that I want to see is our sport going under due to lack of interest...I don't know what I'd do without my Sunday morning PBA telecasts and the prospect of being a PBA Tour bowler. So I find it important to recruit as much as possible, get more people in leagues, keep centers open and keep the sport alive. I think turning away open bowlers during league play would be detrimental in the long run.
However, it is horribly annoying when you are shooting a good game. When you're up on the approach, you take your first two steps and a ball wielding 3 year old screams past you and chucks their ball onto their bumper assisted lane and your shot goes all to hell.
Playing devil's advocate on my own thoughts, one could argue that having the 3 year old next to you is good practice to get you used to bowling with distractions. Or maybe to improve your control. If you see the kid run up, stop mid-swing, walk back, reset (the parents usually see this and hopefully tell their kid to calm down.) I used to have a coach who'd scare me mid-swing and helped me develop the ability to stop anywhere during my swing (for the most part) if needed, which has been helpful.
But at this point, the way things are evolving with bowling, we, as league bowlers, might have no other choice but adapt to the change. As stated earlier, as least we have a league to bowl in. I think we need to learn to adapt...as much as we don't want to...but at this point, I don't think we really have a choice.