The second ability would go infinite with artifact activated abilities that let you take extra turns, like time sieve or magistrate’s scepter as you are presumably skipping the activation cost.
If your intent is that you still need to pay the activation cost but you can just use an opponent’s artifact (making it fairly useless to use on your own artifacts), you may want to make that clear in parentheses. Something like (you must still pay any activation cost for that ability).
[Image of a technological-looking key , held in the hand of an inventor .]
Artifact
T: Untap target Gate.
o4, T: Activate the activated ability of target artifact. Tap it.
-Irini Sengir
If your intent is that you still need to pay the activation cost but you can just use an opponent’s artifact (making it fairly useless to use on your own artifacts), you may want to make that clear in parentheses. Something like (you must still pay any activation cost for that ability).
that·s fair
-Irini Sengir