Calculation of the system states and the individual wire states in a quantum circuit

I am bit confused with calculating the overall state of a quantum gate and the individual wire states.

For example, lets say there are two Qubits, where Q1 is in $frac{1}{sqrt{2}}(vert 0rangle + vert 1rangle)$ state and Q2 is in $vert 0rangle$ state. Then we have CNOT gate controlled by Q1 on Q2 followed by a Hadamard gate at the Q1 gate.

Just after the CNOT gate the total state of the system is $frac{1}{sqrt{2}}(vert 00rangle+vert 11rangle)$.

Then applying the Hadamard gate gives us: $frac{1}{2}(vert 00rangle+vert 10rangle+ vert 01rangle – vert 11rangle)$. As you can see, we get four possibilities of states.

What if I perform the calculations on individual wires? i.e.

1. We perform Hadamard on the first Q1 bit, $frac{1}{sqrt{2}}(vert 0rangle + vert 1rangle)$ state which gives us $vert 0rangle$ state
2. Then for the Q2, apply CNOT which gives $vert 0rangle$
3. Then we calculate the overall state which is $vert 00rangle$

But then we do not get the same answer as before. Are we allowed to calculate like this or am I doing something wrong here?

Thanks!