So you depend on the client to be able to log your server errors?

One of the methods for monitoring for faults is for the client's browser to quietly notify by way of processing a <script> directive, yes.

It should be noted that I have many sites (for clients) running on many shared servers to which I do not have other access and which may or may not be operated by competent server administrators who have randomly broken things in the past.

Any error class that prevents rendering the page will never be logged then.

The error and uncaught exception handler do the appropriate forced closing of open HTML, and in the case that output has not started produce a fully formed HTML response.

In the case that PHP segfaults, or PHP-FPM fails, or... the Apache ErrorDocument's take care of such notification,

And what about clients with script or ad blockers?

An error of consequence is extremely unlikely to be triggered by one person only, the percentage of people who have script blockers is so small as to be zero for all intents and purposes, and I am not aware of adblockers, which I use, blocking the <script> call in this case.