I have escalated this issue to Thrive Leads and they were able to solve the initial issue which is “No Mailing List Shown In Thrive Leads Lightbox After Connecting Mautic”
In Campaigns > I created a contact source as “Contact Segments” and selected a segment a created in “Segments” section.
Inside Thrive Lightbox, I linked my Mautic campaign using API Integration.
I opted-in at 9:42am but I received the 1st follow up email at 9:50am so it wasn’t “immediately.” 7 minutes delay is I think not too bad. Then, I received the 2nd email the following day and the 3rd email next so it is working in order.