I would probably use an ACF field stored as part of the post rather than my own custom field in a widget. I’d probably also remove or hide the standard WP tag box and add that as a custom field as well. Both of these fields would be taxonomy field for tags. One of the fields would save and load terms while the other does not. Then when you work on the front end I’d use the values saved in the one that does not save and load terms to do my displaying of tags.
The main reason that I use ACF is to avoid custom coding custom fields in the WP admin.
Welcome to the Advanced Custom Fields community forum.
Browse through ideas, snippets of code, questions and answers between fellow ACF users