WP Toolkit retrieves actual data about each registered WordPress each day. Usually, this operation takes a few seconds, but in some circumstances (in a wide range, from malfunctioning plugins to real malware), it can take much longer. In this case, the website will be marked as quarantined (which means WP Toolkit will not communicate with this website further) to isolate the impact of possible malicious code on the server.
You are experiencing the same symptoms while checking the integrity. I wonder if it really helps to reinstall the core files. You can check it by running the refresh website data manually right after reinstalling the core (which means not waiting while it is executed by the daily task). Will it succeed or run into a timeout?
Also, it is possible that the quarantined state was initiated by attempting to update your website. Update operation is more time-consuming than just requesting website data, and the timeout for this operation is much longer (30 minutes). Do you configure automated updates for affected websites?
In case your website, in fact, works as expected but slower than usual (but you are confident that it is expected), you can increase timeouts in the configuration file (panel.ini).