I never liked hearing 'styles' divided up into categories like the ones above because, for me, it never made sense to do that. No one who developed a set of moves to work on ever said 'Before I do any testing or even theorizing, I only want to use body movements that you use a snapping motion with/ throw a person with/ grapple and pull on their limbs or joints with/ yada yada yada.'
When whoever started developing a particular training regiment, they observed and theorized on what could work and what couldn't. Then they practiced it to see if it did. Plus, even after they incorporated a certain movement or set of moments into their personal repertoire, it didn't change the physics or bio-mechanics behind it.
I think that is a very modern western view of how fighting styles developed. I'm sure a few early systems did come about like that as well as lots of more modern styles, but many arts were based in social, religious and cultural ideologies and beliefs. A style then should be looked at as the embodiment of an idea, of a philosophy.
Taking a mish-mash approach with no limits as to what you include may sound like a cool idea, but I think many martial artists are making a fundamental error in trying to view MA in this way.
I think many MAists are far too focussed on techniques and look at martial arts as just large catalogues of techniques that you learn and throw together as you see fit in a fight. At this level mish-mash martial arts is fine, but this fails to answer the most important question of any conflict: how do I win?
The mish-mash is like giving 100 men an array of weapons and expecting them to win a battle. They might pull it off, but the chances are it won't be efficient and they will take heavy losses. In my view martial arts are much more about how you win the battle than the weapons you use to fight with. This is where styles become important. The style is the philosophy of conflict and its resolution. The style is your road map to winning the fight. When you forget about styles you are effectively starting from scratch and having to create the strategies that bring victory through trial and error. Furthermore by adding disparate techniques that have no stylistic commonality, you have effectively clogged up your mind with junk.
Its like taking six different jigsaw puzzles and jumbling the pieces in a bag then trying to make a single clear image.
We can do without styles, but if we take time to understand what they truly are the question we end up asking is why would you want to get rid of them?