Wouldn't it be more easily grokked and shorter as well as easier to avoid players messing with to just use something as reminder text like "This card is not fossil as long as it's not [your|a] fossil turn."

Imagine someone trying to pay the cost of a end-of-turn Mana Leak with this, but being unable because the opponent waited with their spell until after the trigger resolved and made the permanent disappear. A static ability avoids that kind of fiddly gameplay tricks for more intended interactions.

Does this untap twice a turn cycle? That seems to be worth more than .

I guess it could be a drop.

Your suggestion is simpler. I wanted my reminder text to be clear as to how the ability was happening.