Can you clarify whether the ideal goal is to focus solely on the sensation of the breath (blocking other thoughts), or just to have your attention on that even if other thoughts are going by in the background?

The latter. I tried to make this clear with my use of "see through" in the post:

After practicing for some time (hours, days, or months) you should be able to "see through" arising thoughts and emotions without getting "stuck to" them. Demanding thoughts and emotions will arise, and by "seeing through" them you maintain your observation of the breath as they continue (unattended to) in your peripheral awareness.

I may have overestimated how long it takes people to be able to practice "seeing through".