Re. Keyboard commands for navigation - tab key to move cursor to next measure for chord symbol entry doesn't work

There seems to be a mismatch there. You are reading the handbook for version 3 but using version 4 of the software.

Do these (version 4) instructions work? https://musescore.org/en/handbook/4/default-keyboard-shortcuts#chord-sy…

You wrote:
Tab move cursor to next measure doesn't work when I am typing in chord symbols.

Use Ctrl + → for "Move cursor to next measure". (Cmd + → for Mac)

BTW:
The navigation commands listed here:
https://musescore.org/en/handbook/4/chord-symbols#commands
...show Ctrl+→ for "Move cursor to next measure". (Cmd+→ for Mac)
and that...
"Move cursor to previous measure" is Ctrl+←

Those commands are not shown here:
https://musescore.org/en/handbook/4/default-keyboard-shortcuts#chord-sy…
Neither are they shown here:
https://musescore.org/en/handbook/4/all-keyboard-shortcuts#chord-symbol…