Why consider Linear Time-Invariant systems?
is there a more compelling reason to consider a system as LTI while
  looking at problems in this field?


What makes the analysis of LTI systems attractive are the following:

Linearity:


If \$y_1(t)\$ is the output due to the input \$x_1(t)\$ and
\$y_2(t)\$ is the output due to the input \$x_2(t)\$ then
\$y = ay_1(t) + by_2(t)\$ is the output due to the input \$x = ax_1(t) +
   bx_2(t) \$.


Time (or shift) invariance:


If \$h(t)\$ is the output due to the input \$\delta(t)\$ then
\$h(t - \tau)\$ is the output due to the input \$\delta(t - \tau)\$.

We then call \$h(t)\$ the impulse response of the system.


If and only if the above are true of a system do we have:

\$y(t) = h(t) * x(t) = \int_{-\infty}^{+\infty}h(t-\tau)x(\tau)d\tau\$

and

\$Y(s) = H(s) X(s) \$

Now, no real LTI system is truly LTI but are effectively so and thus we may use the above "tricks" to analyze them.