Ok, I would agree with your agent and also suggest to skip in July in general.
Unfortunately the towns are quite small and summer is peak tourist season, the place gets overrun.
However, you can still choose to go, knowing this.... I strongly discourage a day trip. I think it would be "easy" to do from Florence, but it would be a very LONG and tiring day surrounded by lots of people.
If you have your heart set on going, I have some suggestions on how to go about seeing the towns.
First, I would highly recommend spending a night or two there, so that you can SEE and EXPERIENCE the towns once the tourist hordes are gone for the day. Many arrive for day trips and then by early evening are gone. That's the only way I'd recommend you seeing CT in the summer.
Second, I would suggest not going between the towns during the day and instead focus on doing something else. In CT, the something else is hiking. I suggest heading up into the hills over the towns and hike during the day, enjoying the sun and nature and come down into the towns that way. Avoid the trains, avoid the main trail along the coast which gets a lot of traffic. Of course, this is how it usually is pre-pandemic, I am not sure how things will be this summer. We will have to wait and see. In any case, in the afternoon and evening you are back in the towns, then you can use the trains and low trails to get between some of the towns (the easy and short Via dell'Amore between Riomaggiore and Manarola is closed because of landslides and isn't planned to reopen until July 2024!). There is another short trail in between them that has you go up the hill between the two towns and isn't too hard.
Consider though it is July and you need appropriate clothes for hiking in hot temps, with hardly no tree cover along the coast unless you go up to the high trails. So wide brim hat is a must and so are proper hikinh shoes, sunscreen, water and sticks are definitely useful.
If you're not a hiker at all, the alternative is the beach. Plan to stay at the beach for most of your day and/or use the ferry service to tour between the towns, maybe head to Porto Venere as it is further south and a bit harder to get to, there are less crowds.
Now as far as whether to do it right away the day after you arrive: I would say no. If you land in Florence, I suggest first see and enjoy Florence. What else do you have on your itinerary? You could stick CT after a few days in Florence before heading elsewhere or even at the end of your time here, right before heading home. The train trip to get there still takes over 2 hours so you have to plan accordingly. If you stay there, then you don't have to worry about luggage, that's one plus

.