I think your itinerary is rushed, but doable.

Yes, Venice is "touristy" but so are all the cities you will be seeing. These are popular spots.

But by limiting your stay in Venice to only 2 nights (which really means ONE full day of sightseeing), you are basically limiting yourself to only seeing the most touristy parts. If you add a couple of nights you can get off the "beaten track" a bit and see the less touristy parts of Venice.

I'd drop Florence and add nights to Venice and Rome. Either way, you don't really have time for day trips.