So I wanted to set N_y (which is just Nsin(theta)) equal to mg, but you're saying it should be N=mgsin(theta). And that makes sense, because the other component of the weight isn't doing anything to the Normal force. I can also see how that's analogous to the other problem with the vertical circle. Then I should be able to get the rest of it from there.
Thanks for your help!
No problem. It took me more than I would care to say to find out which component of the weight you wanted to compensate...
Note that if you use the "centripetal force" analysis, you aren't actually compensating anything. You are finding the value of y for which the weight plus the normal reaction of the surface equate the required centripetal force to sustain the circular motion.
The reason why, when you look at it backwards (or using centrifugal force analysis instead), you are trying to compensate the projection of the weight on the local surface of the bowl is because the projection of the weight on the normal of the surface of the bowl is always compensated by the reaction of the surface, so you don't care about it.
... I don't know if that makes anything clearer, or quite the opposite.
EDIT: I've reread your analysis. Yes, you want N_y = mg to force the sum of the weight and the normal reaction to be horizontal. You were forgetting the principal characteristic of the normal reaction, which is to compensate all forces in the direction of the normal to the surface. Since here the only other force is the weight, this is what gives the relation |N| = mg*sin(theta).
I'm pretty sure centrifugal force is a real thing.
One thing I've learnt during my short existence on this Earth is how irredeemably subjective the word "real" is.