What are the advantages of LTI ( Linear Time Invariant ) systems over other systems?
It's important to realize that in practice many types of systems are used, and only some of them can be regarded as (approximately) LTI. The (didactical) advantage of treating LTI systems in a basic signals and systems course is the elegance and relative simplicity of the underlying theory. Stability and causality are easily checked, and the input-output relation is conveniently described by convolution (in the time domain) or multiplication (in the frequency domain). The Fourier transform is a powerful tool for analyzing LTI systems.

Another advantage is the relative ease with which such systems can be designed. Just think of the vast number of filter design methods.

Even though LTI systems are often used in practice, you will find many situations where other systems are needed. A simple example of a linear but time-varying system is a modulator, which multiplies a signal with a given function. A decision circuit in a receiver is a non-linear system. An adaptive equalizer is a time-varying and non-linear system, since the adaptation of the coefficients depends on the input signal. It would be easy to continue this list of practical examples of non-LTI systems. You will probably learn about such systems in more advanced signal processing courses. However, a deep understanding of LTI system theory will help in grasping these more advanced concepts.