Dulciusexasperis--Actually, the word "independent" and "package" can be used differently. Sometimes a package is just airfare & hotel, or airfare & rental car, which is called "independent" at least at Gate1. Gate 1 calls a "package" as guided tours where every day is set up for you and you go with a group throughout your vacation. The ONLY time I have ever done a package that included airfare and a few nights of hotels is when the price was cheaper than if I would have booked airfare alone. Generally we use miles/points to fly but sometimes a package makes so much sense we use money. Also, in our case for this upcoming trip we are organizing for family members who don't have any miles.
For your next trip, you should look into Delta Vacations, AAVacations, Gate1, etc. Like in our case, Delta Vacations was much cheaper for the airfare alone plus the 4 nights in Paris are free. I mean, the cheapest airfare I could find was $1500 RT from Nashville, so for $1201 we get the airfare and 4 nights in Paris. The rest of our trip is on EasyJet, Luftansa and AirFrance to Munich, Rome and London staying at Airbnbs along the way that I booked without Delta Vacations. All of those other flights within Europe are approximately $60 per person per segment in economy with no checked bag. The trip for 4 of our group is 16 days but my husband and I are staying 4 months in Europe.
There are threads on FlyerTalk regarding using airline vacation packages just to get the cheaper airfare. It doesn't matter what airport you or anyone in your group is flying from, sometimes it's a better deal using one of these packages than just booking airfare alone. It takes a lot of research and if saving money isn't your top priority it's probably not worth the effort for you.
As far as what you seem to be considering independent travel, we do that most of the time. In 2010 we spent 5 months traveling around the world by plane, boat, bus, ferry and train just showing up in towns and wandering around until we found a cheap hostel or guest house. Sometimes we'd arrive by train or ferry and people would great passengers getting off the train or bus to take us to their house to stay in a spare room for $10 per night. Nowadays we usually pre-book our transportation and accommodations before we leave on our trip.