I am not wrong about what is actually written in the rules.
Yeah you are.
This is the RAW:
1) In order to become Hidden in combat you generally must use the Hide action on your turn. You must first become 'unable to be seen clearly' and then you use the Hide action to make a Stealth check (representing you being quiet and concealing signs of your passage). This goes for a creature behind total cover, in a pool of total darkness, who is invisible or even a Halfling standing behind a Medium or larger sized creature. Once 'hidden' you use the rules for unseen targets in the sidebar.
2) You reveal yourself (and cease being Hidden) when you make an attack (hit or miss, unless you have the Skulker feat), after that attack is resolved or otherwise do something to reveal yourself (yell out etc).
3) In order to become hidden again, GOTO 1.
That's all RAW.
Now there might be times when a DM can rule that the above procedure is not needed. Outliers (an invisible assassin in a silence spell, 100' away from the battlefield, on a plain stone floor). That's what DMs do.
But the above 3 points can all be clearly inferred from the RAW.
that they would be able to continue to automatically track position of that foe for several turns when they are running around,
The foe is not making any efforts to conceal their position over those 'several' turns.
They COULD choose to Hide on any of those turns. Instead they're choosing to not be hidden and to instead Attack, Dash, Cast spells etc with their action.
If they want to be hidden, Hide!
I could be down with a creature who has spent several turns invisible and not really doing much (watching the battle, and moving about slowly for those turns) while his opponent was distracted fighting other things being hidden without the Hide action though. It depends on the circumstances.