Buttons set their value to True only in the instant they are clicked . After that, their value goes back to False.
In your use case you would be advised to use st.radio. Radio buttons lay out vertically, so that might not suit you. So, instead use st.checkbox in multiple columns. Then set and hold the value state of the checkboxes in st.session_state.
See “multi-page-app-with-session-state” for inspiration. If you want the page to reset, then use st.experimental_rerun to run the app from the top but still preserving the set states.
HTH,
Arvindra