Collecting All Clients logs every night
You can use an automated central logging system, like Filebeat + ElasticSearch. The automated collection of logs you are referring to could be CMD+ALT+CONTROL+SHIFT+. which is a sysdiagnose command. It takes a long time to run and isn't generally considered a good idea to use as 'standard' log collection.

Using something that is actually made for collecting logs will help you out a whole lot better than a simple script.

If you wanted to go the script route anyway, you can create an applescript that collects to logs and copies it to a server. You can then make a Calendar appointment which has the option to add a script as an action. Another method would be using osascript from the commandline in combination with a crontab entry.

A third more user-friendly option for situations where you don't have a proper server system where you can add these things is Apple Remote Desktop Administrator. You buy it from the App Store and then you can setup periodic jobs to gather logs. You will also be able to completely remotely manage all the Macs.