Reasonable? I guess that's in the pocketbook of the beholder. You definitely can save money by planning farther ahead, especially for airfare and trains. And you are going in high season. So considering the timing, I'd say it's not unexpected.
One time I read that a typical per person per day cost for adults is $150 - $180 -- not counting airfare. I think if you do the math, your estimated costs without airfare for 21 days are pretty close to that. I didn't count the 10-year old. I did 21 days X 3 people X $150 = $9450 + $7000 for airfare = $16,450. RS gave the $150 - $180 pppd number but of course, I can never find that little tidbit when I try to look for it. I do know that range was spot on for my husband and I, depending on the country.
I think the $1,700 for trains also seems high. I'm assuming you are looking at the point-to-point ticket map. I hope you're not planning to use Eurail passes. I think (someone will correct me if I'm wrong) the map has full-price walk-up ticket prices.
If you can plan ahead more, you can save a lot of money by buying early, but you will be committed to the train and time you pay for.
For example, using Trenitalia I tested 3 adults, 1 child (at 10, there is a 50% discount, but I didn't go far enough to find out about all the options), going 5 April from Roma Termini to Firenze S. M. Novella, leaving after 10:00 on a Freccia (arrow) fast train. The price was 122,70 Euro for all of you. The Regionale price was 117,00 Euro, but the journey takes 3+ hours and has train changes.
Skip ahead to 26 June and the Freccia price drops to 69,70 Euro. There are no Regionale trains loaded yet. That's a huge savings if you can plan ahead. Based on clicking on the I for information, I got all excited about the Familia and Bimbi Gratis options, but then I noticed they don't seem to apply to any of the Freccia trains. You might be able to take advantage of those options on other trains.
I adore the Freccia trains. There's just something about that screen showing that we're going 250 kph that fascinates me. But they don't go everywhere.
BTW, I tested the trip from Venice to Munich for 10 July using DB Bahn. Trenitalia would only go to Innsbruck. I have been on that route through the Dolomites. It is absolutely stunning, but the total journey with no train changes takes about 7 hours. As with Trenitalia, the cost for the 4 of you (kids are kids ages 6-14, so your teenager might qualify), dropped from 159,80 Euro for 5 April to 79,80 Euro for 10 July for the exact same train. You can buy a ticket for this train from DB Bahn.
If you do some testing on DB Bahn yourself, be sure to click on Show Details and click on Intermediate Stops and especially on Map View. It shows the route of the train. In fact, you can use DB Bahn for any train schedule or route in Italy and do the same thing. You just can't buy your Italian train tickets from them. I love seeing the maps and dreaming about what I'll see along the way.
However you do this, y'all are going to have a great time.