Understanding why use Laplace transforms for circuits
For the domain of circuit analysis the use of laplace transforms allows us to solve the differential equations that represent these circuits through the application of simple rules and algebraic processes instead of more complex mathematical techniques.  It also gives insight into circuit behaviour.

There are many different transform possible and s domain analysis is useful and perhaps easier to to teach that other techniques often before the full use of complex analysis is embraced.  It doesn't make it any less powerful or useful, just well suited to the problem at hand.  Often laplace transforms are taught even before differential equations are fully embraced so it is a complementary approach.