How to explain time dilation without light clock

If speed of light has to be constant for all observers is the only extra assumption in Special relativity, why can't we explain Time dilation without the light clock?

Sure, it is easy, but less physical and more abstract. Start with the spacetime metric in an inertial frame where $dtau=0$ is the invariance of c postulate:

$$c^2 dtau^2= c^2 dt^2 -left( dx^2 + dy^2 +dz^2 right)$$

Then it is a simple bit of algebra to obtain

$$frac{dtau^2}{dt^2} = 1 - frac{1}{c^2}left( frac{dx^2}{dt^2} + frac{dy^2}{dt^2}+frac{dz^2}{dt^2} right)=1-frac{v^2}{c^2}$$ $$frac{dtau}{dt}=frac{1}{gamma}=sqrt{1-frac{v^2}{c^2}}$$

Imagine a light bulb that flashes once in every $T$ seconds for an observer stationary to the bulb. Can someone explain, just using the fact that speed of light has to be constant, How a moving observer would observe these flashes happening at a different rate $T'$

Yes, in fact this is essentially the starting point for H Bondi's famous "k-calculus" approach. Some people feel the Bondi's approach is more intuitive. Here is an introduction

https://en.wikipedia.org/wiki/Bondi_k-calculus

The light clock is common, but it is not essential. It is straight forward to analyze and very concrete, so it is often used, but it is not essential. If you don't like it then you are free to use something more abstract but still physical like Bondi's approach or an algebraic approach based on the spacetime metric or any other approach you like.