I actually like the sound of this. Seems like on top of learning fluidity and movement, it would be a little more fun than throwing the same set of punches over and over (though, there is probably value to that).
Perhaps I will change it up and utilize your more fluid approach some days and take a more formalized approach on others. If that is a bad idea, I hope someone lets me know that

.
It is never a bad idea not can anyone really prove that. It is just a different approach that places the importance of fighting on what a human actually does in space rather than just punching technique. Remember the one who wins is the one who generally hits more. That does NOT imply lack power but the one who can move in, set up , make better adjustments and has more time in the water learning to swim. Think of bag work like shadow boxing but with contact requiring adjustments. Everything one can do in shadow boxing, one can do on a bag but with more learning about yourself.
The only peopl who will argue this are the ones who do not get the delicacy of which I am speaking and rather choose to see some shit head using bad form and pounding like a caveman on the bag.
Even still, your notion of relaxing on a technique driven mechanical approach to one which is more balanced can never be wrong. The bottom line is that since the dawn of man, we have intrinsicly all learned how to hit each other. Hitting with technique and power is way easier than the intracicies of movement and improvised hitting/defense = fluidity.
If you have a friend and or sparring partner, a great drill that I use when someone learns basic movement and really basic strikes is to have them do the following drill.
When I have taught newer fighters they all know how to punch but hardly know how to move with fluidity or improvise attacks. I have been to too many gyms where people learn punches, start sparring and get stymied and quit. When I work with a class or project fighter, the first few classes/sessions are always about moving. My favorite tool is to take a jump rope, place a cut piece of pool noodle over the handle on one end and make big swinging motions at a student. The have to stay in a fighting stance while moving their feet or bobbing/weaving. In kickboxing, I have them lift a foot as in a check. As they get better, I have them add a jab after they successfully slip it or move. Then they add a second counter. At first it is slow and then over time we pick up speed so their movement makes it harder for me to react to them.
It is fun, challenging (depending on fighter level and swinging the rope) and it places the emphasis on the beginnings of partitioning the brain to move and keep position to fight with good form.
There is more than just one way.