Usually in texts that I've gone through so far, examples of using the path integral approach are limited to the free particle propagator and to that of a quantum harmonic oscillator. It seems to me that for these cases, the path integral approach is more tedious calculations wise than the alternative which is to solve the Schrodinger ODE, like we do in an intro QM course. Are there examples of simple systems where the path integral approach is, so to speak, much easier?
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Are there examples of simple systems where the path integral approach is, so to speak, much easier?
I suspect that for "simple systems" the path integral is always a more difficult approach than a "simple approach" (the simplicity of which is the likely reason for qualifying the system as "simple" in the first place.) In this sense the value of path integral is largely theoretical and/or methodological.
The situation however drastically changes when we deal with systems not so simple - notable the many-body systems. Here path integral (or other "complicated" methods, like Feynman-Dyson expansion or renormalization group) provide systematic ways of performing approximations and/or studying the properties of these systems. On the other hand, Schrödinger equation quickly become unwieldy for more than one particle.
Approximated approaches in one-particle systems might be also facilitated by the use of path integrals - notably the quasi classical approximation (which @Qmechanic was likely already alluding to when mentioning instanton calculations.)
Roger V.
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One powerful application of path integrals is to find large-order perturbation series for the ground state energy of the TISE via instanton calculations, cf. e.g. my Phys.SE answer here. In constrast, large perturbation orders can be computationally hard to achieve via standard Rayleigh–Schrödinger (RS) perturbation theory.
Qmechanic
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