HI
I took those flights recent recently and they were quite full as the rest of the schedule I believe.
It Isn't a big harsh to attend the test as you can do it for free in the airport by presenting the boarding card.
it takes around 20 minutes to get the result and off you go at the gate. This applies from FCO, from LIN isn't mandatory anymore. I don't know much about the JFK FCO route, in pretty sure you only have to produce a negative rapid test result tough.
I wouldn't been worried about..