Would you use analytics tool which helps you to reveal if (and how) your products' website visitors understand what is your product about?

Basically any software which allows you to track conversions with the help of funnels. You can do a very basic version in Google Analytics as well. Another thing that can be applied is using Hotjar and other heat map tracking and user session recording softwares.

https://www.hotjar.com/
https://mouseflow.com/
https://www.freshworks.com/marketing-automation/conversion-rate-optimization/
https://www.crazyegg.com/

I already do! It's called keen.io + hotjar

Large companies do panels and research. I remember many companies offer insights and user session recording as a tool, some of them used to offer human review itself as well.

Then there are optimizers and a/b test tools you could use to test hypothesis without actually taking to the users.

A word of caution is that unless you’re coming from usability background you may build a tool people won’t use. Maybe try to offer it as a “productised service” first and get to interact with clients and understand their needs better before building a tool.

Sounds useful. Although I don't understand how you could possibly tap into the brain of people, to understand how they understand, using an analytics tool. Sorry for all the understands :D