Super-simplistic seems like creating a form with a long-text control that allows notes to accumulate inside a single field. If the contents aggregate and editing them into a readable narrative is important, that minimalism might fully suit the need.
But perhaps you need something more structured? REDCap's repeating INSTANCES can do wonderful stuff. Repeating instances even give you the requested [+add new] button. They can be used fairly simply for purposes like logging contacts. You need to create your contact form, then [pipe] its fields (or any other useful fields in the project) into the "Custom label for repeating instruments" (bottom screenshot).
When you have a view focused on a single subject record, whatever repeating instance forms you have defined appear as panels below the record's grid of forms and events. (I do not believe REDCap offers a way to alter the position or shape of the repeating instance panels.)
You could create a minimalist user dashboard to hide unrelated forms and thus avoid having to scroll to the bottom of the screen to review your contacts.
Repeating instances are actually very well worked out in REDCap now. Our site is running v14.0.14, but as I recall, repeating instances have been maturing since at least v10.
Best regards,
-Tom