Selling well and apologizing well stem from the same core skill -- empathy.
This weekend I messed up pretty badly on the home front. The details don't matter, other than I was a jerk. A crappy apology would have simply been "I'm sorry for being a jerk" and that would have gotten me nowhere.
A real apology starts with "I'm sorry" and than continues with showing that you understand how your actions impacted the other person. "I made you feel small" or "that was time you were supposed to be relaxing and I made you feel anxious, the opposite of what you were looking for."
This is how the other person knows you are sincere. When you can put yourself in their shoes and articulate their pain, frustration and/or anger.
The same goes with sales.
The number of people who reach out to me and say "you're a coach and I bet you want more clients" or "I bet you'd like more time in your day."
Seriously, no shit, Sherlock. By giving me platitudes, you've shown that you have zero understanding of me. You have no EMPATHY for me.
When selling, it's not enough to say "this is a problem for you, yes?" That's just the first step.
You need to follow with "this is how this problem impacts you and your company" and "these are the implications if you don't solve this problem quickly." If you can't communicate those points, you aren't ready to sell. Go back to customer discovery until you have a better understanding for the prospect.