Struggling for route

Have your agent plan the trip via the way you want, or find another agent.

An agent who is trying to get you to go a way that is easier for him/her to plan, may not be the one to be using

Or perhaps the agent is telling you that most of your three days will be train time -- we did Paris > Mürren > Paris and the first and last days were mostly travel. Very scenic, but long hours on the train.

Marie, Can be done, quite easily.

• Day 1 - Cinque Terre to Varenna (on Lake Como).
  • Monterosso (for example) to Varenna-Esino 4½ hours by train
• Day 2 - Varenna via Bernina Pass to Chur. 5½ hours
• Day 3 - Chur to Paris. 5½ hours

A lot of train, but do-able. Look up all legs on the SBB (Swiss railways) website: https://www.sbb.ch/en/
And get a better Travel Agent.

BTW, it is way too early to be looking for next May. This could be the problem your travel agent is having.
Swiss train times for next year have been published, Italian ones may not be yet.
You want to be buying tickets 2-3 months in advance (Feb-Mar), before that they will not be available.

A competent travel agent should know all this.

In fairness, finding an American travel agent that truly understands European train travel would be like finding a needle in a haystack.

Marie, You can quite easily buy tickets for these trains direct from the rail companies (but not yet).
For Italy, that is Trenitalia ( https://www.trenitalia.com/en.html ), for Varenna-Chur and Chur-Paris you can do that on SBB.
If you buy direct from the rail company, you get a "self-print" ticket, a PDF file e-mailed to you. You then just print it out, get on the train and show it to the ticket inspector on the train.
The rules for children are different for each country, Switzerland is 6-16 half price, I don't recall the rules for Italy or France.

I suggest you read these pages from the "Man in Seat 61" website:

• Italy: https://www.seat61.com/Italy-trains.htm
• Bernina Route: https://www.seat61.com/BerninaExpress.htm
• Switzerland: https://www.seat61.com/Swiss-trains.htm
• Switzerland to Paris: https://www.seat61.com/lyria.htm

Thank everyone for everyone for explaining how things work. Looks like I am a day or two short to see anything well in Switzerland - just tempting to see something of the enroute to Paris - but it does seem like a lot of training and not a lot of doing. I think this was the agents perspective rather than discouraging me suggested I look harder at the routes. What are people’s thoughts on “their” favourite route to Paris overland...from cinque Terre?

I am not sure about "favourite" route, most people just pick the quickest, which is what the rail websites give you.
The best rail website for multi-country route planning is DB (German railways): https://www.bahn.com/en/view/index.shtml
That will only sell tickets to, from or within Germany, it does however have data for nearly all railway companies in Europe, so it is most peoples "go to" for finding international connections (of course your travel agent knows this ☺)

Go to the DB website, enter from: Monterosso (or whichever of the 5 towns you are staying in) to: PARIS (CAPITALS = any station in that city).
Then I just picked next Wednesday as a suitable random date, and time: 08:00. In practice train times change little from day to day. Next May will probably be 90% the same as next Wednesday.
the first 3 options were:

Monterosso depart 08:05, Paris Gare de Lyon arrive 21:37
5 changes, 13h32, via Genova, Milano and Basel (through the Alps)

Monterosso depart 09:41, Paris Gare de Lyon arrive 21:45
4 changes, 12h04, via Genova and Nice (coast route)

Monterosso depart 10:55, Paris Gare de Lyon arrive 22:31
one change only, at Milano, 11h36. Direct through the Alps from Italy to France.

That is the quick, direct all-in-one-day routes. If you want to stop over somewhere, just look up trains Monterosso to wherever and wherever to Paris.
It's your choice

Another the route option is:

Day 1 Monterosso to Lugano, 5 hours with train changes at Levanto and Milano Centrale. There is another route late morning with a single train change at Milano Centrale taking 5 1/2 hours.

Day 2 Lugano to Luzern, 5 1/2 hours on the Gotthard Panorama Express. This takes the older, more scenic route over the Alps. The last leg is a 2 3/4 hour boat ride the length of Vierwaldstaettersee to Luzern.
https://www.sbb.ch/en/leisure-holidays/ideas/offer.html/zuege/gotthard-panorama-express#sbbaaa512

Day 3 Luzern to Paris, 4 3/4 hour with a single train change in Basel.