I would absolutely drive to the airport
- Driving into cities is usually much more stressful than driving to an airport
- Heavy bags simply get dumped onto a trolley, rather than having to be lugged up/down stairs etc. Our bags are at their heaviest on the very last day of the trip
- Why add in an extra round of checking in/out and extra transfer time? (such time is invariably NOT holiday time for me. It's hassle time)
- You've already set aside the time to explore and enjoy Firenze and to do it in a good way. Trust yourselves to do it justice in that 1st leg.
This can to a degree depend on the time of the flight back. If it's 6:15am, then that really can change the dynamic and push towards not even a stay in the nearby city, but a soulless airport hotel
If it's mid-late morning, that really is ideal for driving directly there
If it's later in the day, it can be useful to plan for mini day trip to a nice place where you can happily stroll around, but which is close to the airport, to de-stress the fear of hitting traffic. e.g. Trani did that brilliantly for us when flying out of Bari.
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FWIW our thinking changes in the reverse scenario. Being tired when boarding a plane is one thing (hey we might even get some sleep!), but arriving tired and being faced with a long or stressful drive is something we actively try to avoid. In such situations, we're looking for the simplest/safest of ways to get to the accommodation. It's the only time where we happily take a single night in a hotel.
Hence I like your ordering: on foot / public transport when tired from jet-lag / travelling, using that time to shake it off. Exploring the countryside with the aid of the car should be much better for being done 2nd (and it also helps you get familiar with Italian signage, anarchic as it can be sometimes).