I do not see that "everyone" said to start the trip in Rome. At least two of us pointed out that heading straight to Rome upon landing in Venice, a 4+ hour trip on top of the overseas flight, might not be the best idea ( especially with children).
And I tried to explain the extra cost of train travel the same day of a flight. I suggested the difference between advance discount tickets and same-day tickets might be €100 for a family of four. It turns out is it more like €165. Walk-up tickets on Italo for tomorrow ( what they would pay on arrival at VCE to travel to Rome) would be €315 for a family of four ( child discounts probably not available at the last minute). Advance tickets for March, €151.
It is not a good idea to buy the discounted advance tickets for same-day travel after an overseas flight, as one cannot be sure of the timing. If you want to do it, you need to leave ample time between arrival of the plane and departure of the train. Because of uncertainties about possible late arrival of the plane ( or likelihood?) and time to get through immigration, it is advised to leave around 4 hours or so to protect the investment in non-refundable train tix. But if the plane arrives on time and immigration is a breeze, they are left with a lot of unnecessary hang time before the train, with another 4 hours before reaching Rome. I am picturing a very unhappy 5-year-old at this point.
It sounds like they are open to other destinations in Italy from Venice, apart from Rome. Depending on their interests, another place might work. Florence as suggested is a shorter trip from Venice. The Dolomites would be great, but only if they are interested in snow. Perhaps others can suggest other realistic possibilities for 3 days within a short travel time from Venice. With children, I would want to be in an easily walkable town, not a long distance from Venice. Verona comes to mind, or Florence.