They still exist thanks to people who can't be bothered booking it all themselves or are still duped into believing that a travel agent is always the cheaper way.
I personally haven't booked a package deal for a few years now and have no intention of doing so as a holiday where I stay in one hotel for 2 weeks no longer appeals to me at all. I think that's part of the problem too. As someone else said, many younger people are more adventurous and will put in the effort to go to places which my mum and dad would ahve considered obscure or just of limits in the 80s. While they spent their 20's jetting off to purpose built resorts in Spain or Greece and spending about £30 a week on food and drink, I'm spending mine in the likes of Tallinn, Cape Town or Toronto. These places were somewhat off the radar to most package dealers in those days and with the lack of LCCs, the likes of TCX could charge whatever they wanted to send people off boozing and sunbathing for a fortnight in the Med and rake it in. Low cost short haul and good value Long Haul flights are so abundant now, more people can go further "on their own" so to speak without the need for someone doing it all for them.
I have had a look at a few package deals in recent months just out of interest and all the stuff that used to be part of the package is now often treated as an add on extra, such as a meal on the plane, sitting next to one another, a transfer to/from the resort, even a bag on some package deals isn't included. These kind of things may make people think twice about booking an all inclusive package which actually excludes some basic things.
Add these things on to flight prices which are more expensive than an LCC, flight times which are not always favourable and higher prices in the Eurozone and it's no wonder that the package deal and those who offer them are losing appeal, at least on the short haul destinations.