My question is pretty simple. The transfer function of an LTI system seems rather limited in use since the initial conditions must be zero. What are some practical engineering uses of it, and examples? I understand it's mathematical use, but fail to see where it's utility would really come in handy except in very simple mathematical models.
Why would a linear, time-invariant system require initial conditions to be zero? This is completely incorrect.
A linear, time-invariant system is any system that is linear (no state terms multiplying one another or themselves) and time-invariant, meaning that the coefficients don't change with respect to time.
A simple system would be an RLC circuit. Resistance in a circuit doesn't affect its capacitance or inductance, and the same is true for the other terms. The circuit is linear, and, provided you're not actively tuning the circuit (changing any of those values), the system is time-invariant.
An RLC circuit can be tuned by variable capacitors or variable inductors, and such a tunable circuit is commonly referred to as a "radio".
OR, you could have an RLC circuit that features a voltage proportional to, say, a shaft spinning. If you called the voltage "back EMF", then this RLC circuit is referred to as a "motor".
The motor circuit is probably one of the most-used LTI systems, at least for instruction. You can do all kinds of control theory with motors (see also: robots), but heat transfer, fluid dynamics... a lot of systems can be adequately represented by LTI systems. Those systems that don't exactly meet the LTI criteria can typically be linearized and used as an LTI system. See also: Inverted pendulum control, a.k.a. the Segway or Hoverboard.
Imagine hitting a pendulum by a hammer with same force and same direction. The pendulum's response will be always same, yesterday, today, tommorow, and 1 year after. It means, between your impact force by hammer and pendulum's respone, there will be an unchanged relation independent on time. Here, you can model this system as LTI system. Hammer impact and pendulum's response will be system input and output, respectively. And, the relation between input and output will be a transfer function.