This is cool. I implemented an extremely similar system for my last job. The data was dumped into mongo and then we visualized it with stacked bar graphs.

One thing to consider since you're using a count+total model is that the most interesting timings will often be the 90th or 99th percentile, so by calculating averages you might be missing useful information.

I ran into some issues with the implementation after switching to using an async framework since the code was no longer a series of nested function calls. Since the current best practice is coroutines where this will still work I think it's okay, but you should consider how someone using callbacks might time their code. In my case I was in a hurry so I manually called the equivalent of your __enter__ and __exit__, but it was pretty ugly and left a lot of room for bugs.