I would not train from Amsterdam to Nice.. We took train from Paris to Nice two years ago.. and it was about close to my limit of how much time I would spend on a train.. and yes.. you can bring a picnic and relax etc.. but after about 5 hours I was going a bit nuts... lol The train ride from Paris to Nice is 5 hours and 50 minutes ( so 6 hours ) .. and Amsterdam is in fact 3 hours from Paris.. ( done that ride many times) .. so.. 8 hours in a train.. nope.. fly.
However.. I also think it would be nice to spend time in Paris ( and yes I am biased .. love Paris) and being only 3 hours from Amsterdam makes it pretty easy.. )
I also love Nice.. we spent 5 days there in summer two years ago.. visited Eze and Monaco super easy fast and cheap but buses ( one euro bus fare to Monaco!!) or trains ( 3-4 euros gets you to many places ) however..
If you can swing 7 days extra.. I would think a good compromise is take train to Paris.. spend 3 nights there ( only 2 full days.. which is short ) and then train or fly to Nice.. spend 4 nights there ( that's only three full days..)
Remember the day you move from Paris to Nice.. either by train or flight is a non sightseeing day..
Personally.. maybe make it easier.. spend an extra day or two in Amsterdam,, then fly direct to Nice for last 5 nights.. visiting various places, ( Villefranche, Eze, Monaco, Cannes) are all super easy options.