My hunch is, that if you aren't getting satisfying clicks with the brifter when downshifting, it is probably the brifter. As aicabsolut said, they aren't really serviceable. If her method described is deemed futile, you can often find a used version of the brifter (just a single one) on e-bay, although I would imagine the R700's are more hard to find than other models.

You can also completely disengage the brifter from the handlebar without too much hassle by completely undoing the 5mm allen bolt on the outside of the brifter beneath the hood, even without removing tape...it is getting it re-engaged that I find to be a true p. in the a.