I built the search form to also index PDFs and it works perfectly fine when I test it as a logged-in user. But as soon as I go to a incognito-window and re-test it, it does not find anything..
What could be the reason?
The WPFTS Pro version you are using is very outdated. Please consider upgrading to 2.46.x which is the latest. It has a completely new indexing and searching algorithm and it should solve your issue.
It happens because you have problems with wp-cron tasks or DNS configuration.
If you have your website hosted locally, often the domain name is just recorded in /etc/hosts file. This is normally okay for browser access (and this is why indexing is coming well but slow when your WP admin is open). But PHP can not read /etc/hosts by some reason and thus indexing procedure can not go internally.