Pricing path dependent options (Asians, lookbacks, barriers, Americans) is much harder with Fourier. MC simulations are easier in these cases. Recall the characteristic function only contains information about the terminal stock price $S_T$.
For European style options, Fourier methods are extremely popular; in particular because they apply to a very wide range of stochastic processes (much more than just Black-Scholes, Merton and Heston). In particular exponential Levy processes have a simple characteristic function. Some models introduce however new difficulties (e.g. mutlivalued complex valued functions (Heston trap) or residue calculus).
Another downside is that it first requires one to study Fourier transforms and this may take too much time in a lecture. On a first glance, it also occurs less economically intuitive than tree methods or MC simulations. So, it may be just for pedagogical reasons. After all, Fourier methods are still 'younger' than numerical PDE approaches, simulation and trees. For different asset classes, simulation and trees are popular (e.g. interest rate models whereas Fourier methods are less used there). So, it may also depend on the focus of your lecturer.
Finally, some models simply do not have a characteristic function available, e.g. local volatility models. So, you cannot really apply Fourier methods here. For rough volatility models, one has to approximate the characteristic function.
However, there’s some deep financial interpretation and there are some extensions to apply Fourier methods to Asians and other path dependent options. So, they are a popular tool in 'advanced' option pricing.