Of course some people will play it exclusively.
what would that mean for you? You still want to beat the timer, right? you want to upgrade your key right? you want the extra rewards, right?
offering optional timer would do nothing for the pusher playerbase as they will still push as high as they can get.
the only positive they get, is they can try again if they didn’t meet the time once.
I’m even going as far as to say don’t give score if not in time.