Can use of Fourier transform be minimized completely with the help of Laplace and Z transform?
The short answer is yes, if you have the Laplace or Z-transform of a function you do not need the Fourier transform. 

This is because the CFT is a special case of the Laplace transform and the DTFT is a special case of the Z transform. The Fourier transform is used to find the complex sinusoids that compose a function, whereas the Laplace transform finds all complex exponentials that compose a function. 

Knowing this, it is simple to derive the Fourier transform from the Laplace transform. The Laplace transform is defined as:

$$
\int_{-\infty}^{\infty}e^{-s}f(t)\,dt\\
s = \alpha+j\omega
$$

To find the Fourier transform from the Laplace transform, all you must do is assert $\alpha = 0$ for all $\omega$ and $t$. This evaluates the Laplace transform along the imaginary axis only, as any component lying directly on the imaginary axis is a complex sinusoid. This leaves us with:
$$
\int_{-\infty}^{\infty}e^{-j\omega}f(t)\,dt\\
$$

Which is in fact the Fourier transform. 

A similar relationship applies between the DTFT and the Z-transform. The Z-transform is defined as:

$$
\sum_{n=-\infty}^\infty x[n]Z^{-n}
$$
Where $Z$ is a complex variable. To find all complex sinusoids in the discrete world we must constrain $Z$ to fall on the unit circle:
$$
Z = e^{j\omega}
$$

Which leaves us with 

$$
\sum_{n=-\infty}^\infty x[n]e^{-j\omega n}
$$
Or the discrete time Fourier transform. 

The reason the Fourier transform exists is because there are a lot of applications that only need the real frequencies of a function. If only the real frequencies are needed there is no point calculating the rest of the Laplace domain. Why do more work than necessary? 

For more information check out the Wikipedia articles https://en.wikipedia.org/wiki/Discrete-time_Fourier_transform#Relationship_to_the_Z-transform
and https://en.wikipedia.org/wiki/Laplace_transform#Fourier_transform And I would also strongly recommend the text book Linear Systems and Signals by B.P. Lathi for a much more indepth discussion on Laplace, Z, and Fourier transforms.