I think it sounds okay, the sort of addition that probably occurs naturally, since people are always looking for ways to sound cooler and to add more emphasis. Removing the “Let me” (since cut to the chase is used in other situations too) there are quite a few hits for your longer emphasised version, (even outside of English forums like this one) https://www.google.com/search?q=“cut+right+down+to+the+chase”&ie=utf-8&oe=utf-8&client=firefox-b
Hi Timtak, thanks a lot for sharing your view on this question. As of now, when you google the exact phrase “let me cut right down to the chase” there aren’t any results yet (not even this post) so it seems it’s not very popular to put it mildly but the fact that “right down to the chase” already is producing a few results gives me hope At any rate, it’s interesting to read what you think about this so thanks a lot for sharing.