I've been to Rome twice (so far) in the past two years. I love it. The whole city is a walking museum, so you can't go wrong (anywhere.) Yes, a person should dedicate at least two weeks to Rome, if possible. And, even with that, one would never see "all of it." No, lol. Too big, too much.
I read your post, and it seems that seeing Rome, even for a short time, is really in your heart. Do it. Once you reach Rome, the excitement kicks in, and you go into adrenaline drive. As you stated, you won't have your toddler, so it will be easier to move around. You won't have to spend only 51 minutes.
Yes, you can go to Rome from Florence via high speed train. Try to get the earliest one possible. You'll be in Rome just under 2 hours. Get your bearings once you come out of Termini, perhaps grab an espresso and a Cornetto for some quick energy.
Just a reminder, take care to keep your valuables safe -- Passport, Euros, credit/debit cards, phone, camera, etc.-- either use a money belt for the bulk of your "bank stuff" and keep day money close, or maybe in a cross body bag.
I stayed in the Termini area twice, even walked around at night, took buses at night with no issues. Just keep aware with a big city attitude (I don't mean that in a bad way, lol.)
Since you stated you won't be going into most places that you need to reserve and will have long lines, you can see a lot of Rome even if it's only the highlights.
I'm linking the following to (only) give you some ideas https://www.youtube.com/watch?v=Z3pTI9je2ZQ to make the most of your time. Sean, one of the guides, is really fun, down to earth and informative. He's lived in Rome for over 10 years according to one of his videos. I don't know him personally, but, I just like their videos. Maybe browse some of their other videos for ideas.
With a day, it would be good to do a bit of research to map out the sites so you're not backtracking. Below is not in order; just some suggestions.
St. Peter's Basilica is at the top of my list. No entrance fees, but may be a wait in line; security moves along though. If you really want to go in, you could visit the Basilica in an hour or so and return another time if you love it.
Castel Sant'Angelo (close to St. Peter's Square)
Pantheon (Piazza della Rotonda) - beautiful (active) Church
Piazza Navona - beautiful with all the fountains
Campo de' Fiori - fun marketplace by day and lively at night (cute shops around the side streets)
Fontana di Trevi - breathtaking and mesmerizing (and crowded)
Piazza di Spagna - see the boat, read up on the story, climb the stairs to the top to see the Church (Trinità dei Monti)
Colosseo, you could see it from the outside and save the visit for next time.
Monumento Nazionale a Vittorio Emanuele II - (Altare della Patria) - neoclassical architecture, very important and beautiful monument https://en.wikipedia.org/wiki/Altare_della_Patria
For lunch or dinner, try to go away from the tourist ones. There is a restaurant I frequented often (two blocks from Termini) which has good food, family-style, friendly, English-speaking staff, homemade pastas, very inexpensive, open from 12:30 until 11:00 PM. Locals and travelers alike ate there. If interested, I could give you the name of the restaurant.
If you want gelato, and you're inclined to walk to it, (it's not far from Termini) try La Romana. It's very good! Via Venti Settembre 60, 00187 Rome, Italy.
Have fun planning. I think you'll love Rome, and one day, as you want, you'll return.