Not all trains have mandatory reservations on your route. This means that you are always allowed to board. Around Christmas & New Year it is better to make reservations in case, especially for a big group.
Your issue might be that you generally cannot book for more than 6 people in a single booking.
Reservations are cheaper booked elsewhere than on eurail.com. Some advice :
Rome - Florence : oebb.at (13€ mandatory). Always availability on the day, 3-4 trains per hour.
Paris - Amsterdam : https://www.b-europe.com/EN/Booking/Pass#TravelWish (passholder quota, 32€ mandatory). This may be your biggest issue (what's your travel date ?) but there are multiple options via Brussels or Lille. Just let us know.
Amsterdam - Berlin - Prague : oebb.at (3€) or bahn.com (4.90€ with a seat map).
Prague - Vienna : oebb.at or cd.cz (about 3€ optional). Trains operated by CD have a seat map on their website and vice-versa
Vienna - Zurich : oebb.at (3€ optional). Seat map available
Questions welcome. More info : https://www.seat61.com/interrail-and-eurail-reservations.htm