I am not sure how far down the rabbit hole you want to go with this, but it is certainly possible for Windows to identify when the user is idle and display a screen saver. It is also possible to capture when a logout, or shut down event happens in Windows. Further, it should be possible to identify when the web client is being closed.
Doing all of these things would require a service or client of some kind installed and running on each computer to monitor for these events... and then do something about it. One option might be to change the user's status to Away. Add that to the list of requirements and what that requires.
I am fairly confident we could capture the majority, if not all, of the important events. While it might take a week or two to build a prototype client service, taking the prototype and making it a "real" solution is going to add significantly more time (time is money one way or another).
I always have to ask the question -- of all the priorities I have sitting on my desk, is this the most important thing I need to focus on for the next month? Is it worth the effort? I'm going to need to create a service, an installer, and because I have 1000 endpoints, I'm going to need to support automation because I do not want to run around to 1000 PCs every time there is an update, and that includes recognizing when an updated version is available, and then present a "upgrade now" button, and installation guides, etc., etc., etc.
The first questions should be (1) Is it technically possible (yes), (2) how far down the rabbit hole do I want to go with features/polish, and (3) does the effort required justify the necessary investment in time/money?
I performed a quick ChatGPT asking if there was a free utility that could capture the events for you. Maybe not all the desired events, but even Windows Task Scheduler can apparently detect and take actions on idle, logout, or shutdown. If that's good enough, then you just have to figure out how to call the 3CX xAPI to change the user's presence status. Is this approach easily distributable to 1000 computers?