Advice on latest time to attempt travel from Florence to Monterosso

It depends on how you are going to travel from Florence to Monterosso.

By train? If there is a train departing around 17:00 and you are not interested in Florence, why not?

By car? I wouldn't drive in a foreign country, on the mountain roads along the coast of the Cinque Terre after a long flight.

"scheduled to arrive in Florence at 3pm." Is that arriving in Florence (the city), or landing at Florence airport?
If that is your time at Florence station, you can get a 15:00 train, arriving Monterosso 17:38. Or a 15:28 arriving 18:17.

If you mean time at the airport, you have to add time to disembark, passports (depending where you are flying from), possible flight delays, getting into Florence. In which case you are likely to get to Florence station 17:00-18:00. Your trains are then 17:28 arriving 20:38, then 18:28 and hourly up to last train 21:28 arriving 00:22.
So no problem getting to Monterossa, but the 11pm hotel deadline means I would stay in Florence.

Of course we are interested in Florence. We will visit later in our trip.

We are landing in the Florence airport and plan to travel by train.

Arriving in Florence, you will have gone through passport and immigration in another city. IF you don't have any delays, or checked luggage and the travel gods are nice, you could taxi into the station and MAYBE catch the 16:28 train arriving Monterosso 18:23. Lots of "ifs" there.

It's certainly possible to get to Monterosso the same day. The latest train with a pre-11pm arrival in Monterosso departs at 7:28pm, and there are departures from SMN at least hourly before then. All the trains departing SMN at 28 minutes past the hour stop at Rifredi station, which is closer to the airport and a less expense taxi ride, at 34 minutes past the hour. You'll have to decide whether you and your family would be OK with another 3+ hours of travel that day or would prefer to get some rest before continuing on to Monterosso. I'd push on if it were me, but I'd have to think twice about that if flying to Florence after a long overseas flight with younger children.

Go to Trenitalia app and take a screen shot of all the options. As long as you are on the 6:28 train or earlier you should be fine, but tired. The 4:28 train has only 1 connection in Pisa, but all the later trains involve two connections, in Pisa and LaSpezia. Depending on whether you've travelled via train before in Europe, and how many people and ages, might be the determining factor as to staying in Florence or pushing on to Monterosso. While travel by train is easy, it's a bit confusing for rookies to make sure you are on the right platform, get off at the right stop, and get on the right train. The more people you are herding with luggage, might make this more stressful than staying in Florence and taking a train the next morning. And once you get off the train in Monterosso, you have quite a hike, depending on where you are staying. And if it's dark, makes it even harder, especially with luggage.

We did the same a few years ago. Quick taxi ride to train station. Unfortunately we had problems using our credit card (rookie error, knew this was a problem and forgot), had to leave the station to find a Bancomat, and then came back and bought tickets for the next train. We probably lost about an hour doing this. I remember in Pisa, confirming with an elderly Italian woman that we were getting on the train going to LaSpezia. She made sure that we got off at the right stop. Italians are so helpful, even though we spoke very little Italian. Upon arrival in Monterosso, we had a taxi take our luggage to the hotel while we walked. Then we paid the taxi at the hotel.