What I understand from your question is why modelers prefer unconstrained differential equations than constrained differential equations system. Generally speaking, it depends on the nature of the problem and the degree of abstraction a modeler seek. Besides, unconstrained models can be solved easier than constrained ones.

But once we formulate constrained differential equations as unconstrained differential algebraic equation we are bound to develop or use the techniques of solving DAEs which is quite complex task.I think we pay a lot to remove constrained nature of original differential equations.

Usually, the constraints arise automatically in modeling. This is due to conservation laws or interconnection of systems. Also, it is often much easier to derive a model in euclidean coordinates than in generalized coordinates. To see this, consider a simple pendulum without friction. This is easy to model as a differential equation (with generalized coordinates) or as a differential-algebraic equation with euclidean coordinates. However, if you go to an n-pendulum, that is at the end of a pendulum is another pendulum attached and this process is continued until you have n pendulums attached to each other. The modeling process in euclidean coordinates is quite similar to that for one pendulum. However, the modeling process in generalized coordinates is highly difficult, since all tangential forces need to be computed!

Another advantages comes in terms of numerical simulation. If you are interested in preservation of a quantity (usually energy), you can add this constraint to your model (if not already added). Hence, the solver will preserve this quantity (up to the precision of the solver) automatically. If the constrained is resolved/differentiated such that the system can be modeled as an ODE, one needs a symplectic integrator to preserve the quantity of interest.

If you are interested in solving DAEs, I'd like to mention the work by Volker Mehrmann, Lena Scholz, Andreas Steinbrecher (TU Berlin), Peter Kunkel (Leipzig), Stephen Campbell (NC State) and many more.

Absolutely, the constraints normally arise through conservation laws in physics such as conservation of angular momentum implies symmetries among the variables that reduce the state space.  The state space of a rigid body is also reduces because of symmetries.  

These symmetries give rise to a reduction in the dimension of the configuration space and often the resulting configuration space is a Lie Group, e.g. the configuration space of a ridged body is S(3,R) - the three dimensional rotation group.  This gives rise to the Euler angles associated with the system.  

Now the question gets down what is the best method to describe the dynamics? In many cases such as ridged bodies (airplanes, satellites, etc.) it is via the use of the radius and the Euler angles.  

The answer is really problem dependent.  

I wanted to add a reference and hit the add button to fast.  A good discussion of these issues is in the treatises by Abraham and Marsden,

Foundations of Mechanics 

https://www.amazon.com/Foundations-Mechanics-2nd-Ralph-Abraham/dp/080530102X/ref=sr_1_2?ie=UTF8&qid=1493310261&sr=8-2&keywords=foundations+of+mechanics

There seems to be a new addition out from the one I have.

https://www.amazon.com/Foundations-Mechanics-Ralph-Abraham/dp/0201408406/ref=sr_1_3?ie=UTF8&qid=1493310261&sr=8-3&keywords=foundations+of+mechanics