I thought っきり only went with past-tense verbs?
I’m no expert, so take my opinion with a grain of salt, but if I were to hazard a guess, it’s because きり can mean “only” and “since”. When it follows a verb in the past tense, the “since” meaning is what is typically being expressed. In this example, it’s not talking about an action that has already occurred. It’s specifically talking about “the thing that is ONLY leaving child rearing up to parents is bad.” The というのは part is making the entire thing a noun phrase in this case.
But if there is anyone that knowns more about this, please correct me if I’m wrong.
It looks like 任せ is a noun in this case, not a verb.
I’ll also add that It’s not necessarily true that you can only use っきり after た-form of verbs, sometimes you will see it after the masu-stem. However this pattern seems to only be seen with certain set expressions like かかりっきり and つきっきり.
As explained in the video, this usage has the nuance of only doing one thing without doing anything else.