In my experience a sewing pin is just about the perfect thickness to get into an USB-C port. If you're even halfway careful you can dig out all the lint without damaging anything.
I do have to say that USB-C seems to be much more lint-prone than micro- or mini-USB. I have never needed to dig out any lint on my previous phones, but have had to do so a fair few times on my latest phone.
Thanks for sharing your anecdote. Mom & Dad didn't give me a call about your phones not charging yet, but knowing that it's doable is a relief.
I don't remember digging lint out of any previous USB generations either. Only a couple of USB-A connectors, which were integrated to some smaller MP3 players (yes, I remember them!).
I do have to say that USB-C seems to be much more lint-prone than micro- or mini-USB. I have never needed to dig out any lint on my previous phones, but have had to do so a fair few times on my latest phone.