This is a great question. in its almost bold simplicity it highlights the potential circularity in the concept of mass that keeps me up at night. @swansont explanation is perfectly correct but it does not address the deeper question: what does device on the right actually measuring? So far I do not have the answer to this question. But whatever it is - it does not required to solve gravitational systems. Here's runnable colab notebook that demonstrates it: https://colab.research.google.com/github/AntonRize/WILL/blob/main/Colab_Notebooks/R_O_M__MASS_G_FREE_CLOSURE.ipynb
If python is not your jam here's desmos demonstration:
https://www.desmos.com/calculator/mjen4ms452
So the question remains: what does device on the right actually measuring?

1 hour ago, Anton Rize said:

This is a great question. in its almost bold simplicity it highlights the potential circularity in the concept of mass that keeps me up at night. @swansont explanation is perfectly correct but it does not address the deeper question: what does device on the right actually measuring?

Primarily because that was not the question that was asked, making this off-topic, and you’re posting to advertise your speculation, which is against the rules.