You are going to get responses all over the place in regard to this. I would say, do what feels right to you.
I don't feel like you should contact a person via the convo system and ask them to leave a review. However, when a person first purchases, Etsy sends them a receipt, and in that is the "message from the seller". I say thank you for your order along with other stuff. But, I also add that I love to hear feedback and for them to let me know about their experience with Kylee Inspired Crafts through the feedback system if they have some time. That is the one and only time I mention it.
I have had a person contact me personally themselves one time telling me how great their purchase was, and I asked them if they would mind leaving feedback. As you said, they didn't even know about it or how to do it. I am not sure I would do that now, though.
Etsy does remind buyers that they have feedback to leave at the top of the screen if they ever log back on. I know because they have done that to me about leaving feedback before.
Regardless, it is your shop. Do what feels right to you.