Both have an important role in your connections with other people but its useful if you learn to separate them.
Or think about it another way... when you listen to music by Bach, do you feel an emotional connection? Do you see it as crass or alienated, or are you feeling something that a long-dead composer wanted you to feel, and grateful for the experience?
In the context of Dale Carnegie, I think that a genuine connection requires the active involvement of two people interacting with one another. For all that the emotions involved are unquestionably valid, I do not consider the genuine emotion one person feels for an imaginary other person to be a genuine emotional connection with another human being.
Bach, to my knowledge, did not like to rattle on about the importance of synchronous emotional engagement.
It may be possible that the detection heuristics a given individual relies upon might, upon occasion, be a bit less reliable than could be hoped for. I've witnessed both false positives and false negatives.
Again, you're right. People do earnestly and honestly believe in their ability to detect genuineness.
You have proposed an Emotional Turing Test! The price for failure looks high!
It also made me a lot more cynical. I realized around halfway through that the author was (very skillfully) deploying his techniques in the direction of the reader. Further, the author is long-dead. We cannot possibly have a genuine emotional connection.
This helped me understand that genuine empathy is irrelevant. All that matters to convincing people is that they feel empathized with. How you actually feel isn't important, though for many it's likely to be by far the easiest and most reliable way to get there.
Dale Carnegie provided me with a very useful set of tools that I can use to achieve the outcomes I want. For that I'm appreciative.