It is no crazier than some of the things I've done to see art that I want to see.
Here's some thoughts for you.
Is there any way you can get yourself a few hours onto Italy time before you leave? We are retired and can do a full four hours which completely eliminates our jet lag, but we took family to Italy last year and they reported that even a couple hours helps. Move your mealtimes, too, not just your sleep.
Do you know what you want to see at the Uffizi? (I remember how to spell it because it was an OFFICE, so two F's, one Z.) I just spent a whole day there for the second time and found it very useful to download maps from the Uffizi website, print them up, and mark the rooms I wanted to visit. It really helps to strategize. You cannot see everything, no matter what. Do not feel bad about ignoring wonderful art if it's not your favorite and you don't have time. I have now spent a total of 11 weeks in Florence and not seen everything I want to see or see again --- Italy is just like that and Florence in particular is ridiculously stuffed full of art.
Can you possibly book an 8:15am ticket? It will still be crowded, especially around any famous painting, but it gets even worse as the day goes on. You are supposed to show your passport or some ID to prove that your ticket is yours, but they just waved me through --- don't count on that, though.
Here's a tip that may or may not be useful --- there is a cloakroom but it's not for "cloaks"! I was not able to leave my coat and sweater there, and had to carry them all through the Uffizi (which was quite warm). HOWEVER, if I had taken a empty backpack and put my outdoor layers into it, I could have left that in the cloakroom. This will probably only apply to you if it's raining and you have a raincoat.
Do not bother to take photos of the art, unless it's some detail you want to remember or look up about later --- there are plenty of great photos online.
Guides and tours drive me up the wall, but that's just me --- they speed past things I want to see and linger for endless minutes in front of things I don't care about. The DIY approach works best for me. Listen to Rick Steves' audio guide before you go, not during. Maybe a couple of times. There is helpful stuff on Youtube, too. I have learned a lot from Elaine Ruffolo, Rocky Ruggiero, the Khan Academy, the British Institute in Florence, and just doing a search on youtube for Uffizi Gallery.
Yes, there will be more trains from Rome to Florence. Lately, I've been finding Google Maps easier to look at for trains than the Trenitalia website or app. It's also good for buses, if you need to take a bus in Florence.
Good luck with your plan!