Well, we're making progress .... slowly ..and learning our server environment.
So you've made the pinfo page and can see the settings for time for scripts to run etc. Did you try setting those higher ... defaults are normally low ... like 30 or 60.
Granted those are intended for local server processing, but let's remember how having CF in the middle is supposed to work ... all data goes through CF ... across a network you can't control. According to the page, if CF can't get whatever in 100 seconds ... error 524.
In the error logs shared ... remember, I don't run Litespeed so pardon my dumb question ...
lines do say 'info' but I see the loopback IP address - 127.0.0.1 and a port + APVH
with 404 errors .... file not found ... for 'index.php'.
Here's a link to/for Wordpress issues with Litespeed 404's
appears Litespeed uses .htaccess files. Got any in your setup?
If so, what does it contain?
Compare what you have to advice given in link above.
Am not saying that will fix this ... just doing some leg work for you.
Cron job ... you've got it pointed to the cron.php script in /admin/ and that should require you to pass a password with the URL. There is a cron.php in /admin/cli/ you should be using.
So it should be if using curl
curl -k https://yoursite/admin/cli/cron.php
Note that uses the web and pointed to /admin/ the area you can't get tol!!! Am wondering if that somehow gets routed to CF and thus is your issue. Cron jobs could run longer than 100 seconds!!!
However, cron job best if running right in/on server ... and just php talking to the cli/cron.php script. Never leaves your server - internal only ... no networking involved thus NO CF!
LIke so ... and so you know it's running I'd do a redirect to a log file you create:
/usr/bin/php /path/to/moodlecode/admin/cli/cron.php > /var/log/moodle-cron.log
rather than the redirect to 'la-la-land'.
Above assumes your server logs are in /var/log/ if not there adjust.
To create the log file, as root user:
That creates a blank/nothing in it file to which output of the php running of the cron.php script is redirected. Why? If need be one can watch it execute in real time with:
tail -f /var/log/moodle-cron.log
Also /usr/bin/php is a guess ... on a LS system ... you do want the PHP-CLI not PHP used for the web.
Test: /usr/bin/php -v
If you do not see php-cli in output of version, wrong php.
Ok, I've taken this about as far as I can ... don't run litespeed .. don't use cloudflare - so I have NO ... none .. nada .. first hand experience with either.
Best of luck!