Hello Tim L, and welcome to the Forum!
My first impression is that you are probably trying to do too many thing too far apart even for an "almost 3 week" trip. Going from Venice to CT is crossing Italy at nearly it's widest point and I would expect 8+ hours door to door to accomplish this. Anywhere in Tuscany to the Amalfi Coast is probably close to the same amount of time. Travel to AC is logistically difficult and full of potential delays. Also keep in mind that 2 nights is only 1 full day somewhere with days full of packing, lodging and travel on either side. So you'll have one full day in Venice, travel nearly all day to get to CT, have an evening, one full day and then maybe a morning there followed by 3+ hours to Florence getting out of your hotel probably late afternoon at best.
I know there is an urge to go everywhere and see everything but you also need to account for the realities on the ground of doing the country both east/west and north/south all in the same trip. This is especially true when you are including areas with difficult travel logistics.
Dropping CT would allow you to add a day to Venice and a day to Rome (both really deserve it) and eliminate a full day of travel which is almost getting a day back for the effort. My suggestion would be to drop the Amalfi coast, add a night to each Venice, CT and Rome. Venice and Rome both deserve more time and CT deserves at least another day to justify the full day of travel to get there. I think one of the two coastal areas is the outlier but which one is more important is up to you.
If you're determined to do all of this I would look at replacing the Amalfi coast with Sorrento since it is easier to get to and you can do both the AC and Capri as day trips from there. But as I said I think instead you should aim to do less overall. Part of Italian experience - especially in rural Tuscany - to slow down and really start to feel the slower pace of life and the difference in priorities. Planning always means making the hard decisions about what to leave in and what to leave out, but it will ultimately lead to a better vacation if you give yourself time to experience these places.
Hope that helps, have a great trip,
=Tod