It sounds like what you're describing is some form of analytics. And yes, that is a thing.
Not only traditional analytics such as Google Analytics that show you the various journeys people take on websites (what pages they go to, how they arrived, how long they stay...) but also page-level analytics, such as heat-maps (showing you the areas of the page they tend to click on) and other various stats too.
There are many tracking / analytics systems you can embed that can do this for you. Far easier than trying to log everything that happens in a log file, because while you can do that you'd still need to analyse it all, and log files themselves aren't really that user-friendly. And they get big and heavy, very quickly, slowing things down for the user (and yourself).
Here's some examples, such as HotJar, Kissmetrics and CrazyEgg: https://blog.prototypr.io/10-analytics-tools-for-optimizing-ux-886e64a6bcdd
Now, something to bear in mind:
The main issue with analytics such as this is that they only show you what did happen. There's no coverage of what didn't (so, no way to log who didn't visit, or who didn't click on various items) and also doesn't give you any reasoning behind why events did or did not happen. So you'll have to interpret it all yourself. And while the data itself is objective, your interpretation of it certainly isn't. So you should pair this with more traditional usability reviews - namely, talking to people who use it.