I would definitely do train all around
4 adults and 2 children means 2 taxis to get to the airport and then from the other airport into town (could take public transport but depends on amount of bags you carry for 3 weeks vacation AND a family visit)
Trains are super comfortable and less prone to delays in the winter.
Many airports get fogged in during the winter.

Get you train tickets 3 months in advance.
use scnf.com or bahn.com or obb.at/en for the french, german and austrian train companies.
3 months in advance seems to be the magic time when seats open up on many trains and at the cheapest prices. Buy while the euro is low.
closer to the date there are many sales but coming from so far you cannot risk it.
For shorter train rides, you could wait or buy the same day if the difference is not big.
If you can afford it, go with first class trains. not that much more expensive and I think the quiet and space are worth it.