If you want to use the word 'independent' then you really should not be using the word 'package' at the same time. Either you travel 'independently' or you buy a 'package'. You do in fact want to book a 'package' it seems and not travel independently. If you wanted to travel independently, you would book airfares and you would book hotels, independent of each other.

I only travel independently and would see no difficulty in booking airfares from several airports and then booking hotels at the destinations you wanted to stay in. But the idea of booking packages from different locations, all with one package operator and all within the budget of different people sounds to me like a very unlikely thing to be able to do. You have now given an answer you found for travel from Nashville. What about from the other airports you mentioned in your OP? Are those travelling from those other airports also able to get as good a deal as they might with another package provider? And what about your wish to visit Munich and Rome, are they also part of what you are booking with Delta Vacations?