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!).
Instead all the lint collects around a hair-thin port insert which has all the pins on and can be just easily broken while trying to clean the lint.
I've removed tons of lint from my two iPhones for years, and both lightning ports work like they've worked as in day one.