Circuit analysis by local application of Kirchhoff's current law and Ohm's law
There are several problems with the approach you describe.

If there is a voltage source connected to a node, then you can not directly write an expression for the current through that source. In other words, it introduces an unknown current. There are ways to deal with this but they require considering more than one node at a time.

Ohm's Law applies only to ideal resistors. It does not apply to any other kind of component.

You can't arbitrarily assign a value for the voltage across an energy source that is an ideal current source. 

Each node in the circuit must have its own voltage. If you go around assigning 0 voltage to different nodes it will be very difficult to form a set of consistent equations.

So the bottom line answer is no, you can not, in general, solve a circuit analysis by analyzing one node at a time. You must create a set of independent simultaneous linear equations and then solve that set of equations.

There is an approach that relies on linearity and uses something like "the whole is the sum of the parts". The technique is called superposition, and it is much different from what you describe.