The "link-clicking" can be done using a Google App Script. I've used it before to auto-accept AWS opt-in notifications for Elastic Beanstalk environments.
My code was tied to a Google Sheet that would hourly pull matching emails, use a regex to extract the link, send an HTTP request to the URL, and record the URL and response in the spreadsheet.
Having a high level description of the code isn't as useful as the code itself. Alas, my code was part of my Google account at a previous employer.