Train travel from Rome to Amsterdam

I would not use a railpass.
Rome-Florence should be bought well in advance in order to get best price, maybe as low as 9 euro for a nonrefundable nonchangeable ticket. With a railpass, you would need to buy a 10 euro reservation for each person.

Florence-Nice with a combination of regional and intercity trains. Again maybe as low as 9 euro per person. a reservation fee of about 4 euro is needed for reservation on intercity trains using a rail pass.

Nice-Paris can be done with a TGV direct train for around 25 euro per person. Not only does a railpass holder need to buy a reservation, they may not sell a reservation because the limited quota of railpass holders for that train has been reached.

Paris-Amsterdam on the high speed Thalys train requires a reservation fee for pass holders that is as much as and advance purchase nonrefundable, nonchangeable fare.
You can buy all of your tickets at www.capitainetrain.com. They handle all these train companies and have identical prices. Print at home tickets.

It is not likely that you will get these fares for €9; I've never even seen one. Advance fares around €30 are more reasonable to expect. If you're ready to lock in dates and times for each trip, about 3 months out is the time to find the best prices. For Nice-Paris, you can look earlier; a couple of trains per day, called iDTGV, are already for sale through the first week of September.

Passholder reservation prices for Thalys are $30 in 2nd class or $45 in 1st and, as of this year, they are available with any pass that covers France. So if you use a pass, it could be the 2-country France-Italy Select pass for $250 per person in 2nd class or $310 per person in 1st class. Seat reservations on your other legs cost €10/$12 each, with Nice-Paris being the most important one to book ahead. There is no convenience factor since all of your legs are reserved trains.