So, I know there is a way to add notes to a form submission and save them without sending an email to the user who submitted the form. I am wondering is there is a way to display that note on the Entry List page?
Displaying notes in the Entries list is not a feature. An easy workaround would be to add a Paragraph field to your form, with Visibility set to Administrative, that you can edit from the Entry details page to add your note to this field instead of using the Notes section.