That is entirely your decision based on airfare price, convenience, and whether you plan to visit Rome as well. Flying to Florence (which is my preferred airport since it is there I need to go) usually carries an extra cost compared to Rome (which has slightly better prices), but if you don’t need or want to go to Rome, then the airfare savings of flying to Rome may be negated by the extra time and cost of having to take the train from Rome and also spend your last night in Rome (flights to the US leave in the AM).
Flying via London LHR may not work too well to FLR, since British Airways only has one flight a day from LHR to FLR and back. Because of that (my home airport is SFO) I generally choose either AirFrance (via Paris CDG) or Lufthansa/United via Frankfurt or Munich, or Swiss (which is part of Lufthansa) via Zurich. KLM Via Amsterdam is another option. All of the above have 3 to 5 flights to Florence (and Back). I have no issues with AirFrance, just be aware that the Paris airport is huge so you need a couple of hours to connect to the next flight without worry.
I read that you plan to rent a car in Florence. You won’t need or want a car in Florence, so I presume you are going elsewhere in Tuscany, maybe a rural location. If that is the case and you decide to fly to Rome, then you might consider renting a car in Rome and drive the 3 hours to whichever rural location in Tuscany you plan to go. Just make sure you are not too jet lagged to drive when you arrive.