The Minhag by most Shuls where there is no Birchas Kohanim and the Chazan just says the Birchas Kohanim is to say Kein Yehi Ratzon. Why do we say Kein Yehi Ratzon instead of Amein?
I think this is what R'YDK meant in his answer, but am not sure, so will state it separately. Taame Haminhagim 117 says in the name of the Avudraham (or of abu Dirham, supposedly):
A reason we don't answer "amen" when the chazan says birkas kohanim is that we answer "amen" only when we hear from someone blessing. Thus, there are some Jews who answer "ken y'hi ratzon" instead of "amen".
Taame Haminhagim writes in kuntres acharon there in the name of the L'vush what may be the same answer:
A reason we don't say "amen" is that since the chazan says it only in the language of a request, starting with "elokenu... bar'chenu...", the language of "amen" doesn't follow it, rather "ken y'hi ratzon", meaning "we, too, ask of Him yisbarach that He do so, and so should be His will yisbarach. And that's what we practice.
I'm remembering this from a while back and don't currently have the source.
Amen is used following a Bracha or Tefilla (Various interpretations. Simply, to confirm your belief in the statement), but not when you're quoting a beracha. When we substitute "Borcheinu vabracha", we are not stating a bracha- that's the job of the kohanim. We are requesting that Hashem should bless us with those barachos even if there aren't any kohanim to verbalize the brachos. That would make the pesukim a quote of a beracha and not the beracha itself.
I suspect (no source) that we want to differentiate between the actual blessing and the Chazzan's substitute for it, so that no one will think that the Chazzan is invested with the ability to do the true blessing.
When a Cohen Rabbi imparts the blessing, the congregation responds "Amen"; but when a non Cohen Rabbi imparts the blessing, the congregation responds "Ken Y'hi Ratzon" (So, it may be His Will.) Please, notice that there are 3 blessings. So, there are 3 answers.