The access log does not tell any error.
The error log does.
The error log generated by WP by default will be named debug.log
The server error log will NOT be in your WP Install. It will be in your server root level. It usually is named error.log, but, it could be named anything really, just like the WP debug log can be named anything the webmaster or server host wants.
Your server has both, an access and an error log. Always. There is no host without errors, because they log everything, which includes false alarm errors such as when upstream files are requested by hackers that are not there. And that happens daily thus the error log exists, unless your host consciously hides or disables it. At which point I would consider changing hosts.
However, if wp debug does not log any error it is likely a server issue that can’t be read or caught by WP.
Thus, please share the server error log, or ask your host to give it to you (perhaps only the relevant time, so it is not too long.)
The flush error you got there instead is not due to CP or even WP and has nothing to do with the thread here. It is something else, see https://wordpress.org/support/topic/plugin-problem-and-ob_end_flush-error/#post-10499379:
If you search for this error you will see it is a common WordPress error, which pops up when you have zlib output compression enabled. Disable it and it should be solved. For more info, do a search on the error, or see this thread:
@zigpress makes a valid point there. However, I’ve never seen cPanel “thinking” for me unless the host configured it to follow some rules. I’ve used cPanel with partial WP installs, with custom codes, etc, and never had any issue, as long there where no rules saying “do not open file XY” or whatever.
I suggest, at this point, asking the host why the file is not opened or why it throws a 404 (for some) and/or why it throws a 503.
The very least they will be able to tell you “there is a PHP error” or “there is a permission issue”
Since the error is not isolated to one person, we cannot simply write this off as “cPanel fault” or “host fault”. It would be something rather common, since more than one person seems affected.