Regarding the advantages of generalized linear phase filters
Note that with the definition of generalized linear phase $\phi(\omega)$ according to

$$H(e^{j\omega})=|H(e^{j\omega})|e^{j\phi(\omega)}\tag{1}$$

and

$$\phi(\omega)=\alpha\omega+\beta\tag{2}$$

the restriction that the impulse response $h[n]=\text{IDTFT}\{H(e^{j\omega})\}$ be real-valued only allows two possible values for $\beta$:

$$\beta\in\{0,\pi\}\tag{3}$$

This is due to the required conjugate symmetry of $H(e^{j\omega})$: $H(e^{j\omega})=H^*(e^{-j\omega})$. The value $\beta=0$ obviously results in a system without phase distortion, and the value $\beta=\pi$ results in a sign reversal without any other phase distortions.

However, if you define generalized linear phase according to

$$H(e^{j\omega})=A(\omega)e^{j\phi(\omega)}\tag{4}$$

with $\phi(\omega)$ given by $(2)$, and with a real-valued but possibly bipolar amplitude function $A(\omega)$, then for real-valued filters the constant $\beta$ can take on four different values:

$$\beta\in\{0,\pi/2,\pi,3\pi/2\}\tag{5}$$

For $\beta=0$ and $\beta=\pi$ the amplitude function $A(\omega)$ is even, and we get no phase distortion apart from a sign reversal for $\beta=\pi$. However, for $\beta=\pi/2$ and $\beta=3\pi/2$, $A(\omega)$ is odd, and there will be phase distortion, i.e., the group delay does not equal the phase delay. So for narrow-band signals, the envelope experiences a different delay than the carrier. Basically, - apart from the delay - the carrier is just shifted by $\pm 90$ degrees. This is what is desired when implementing differentiators or Hilbert transformers. Consequently, systems with generalized linear phase and with $\beta=\pi/2$ or $\beta=3\pi/2$ are important for implementing causal (FIR) approximations to differentiators and Hilbert transformers, where the phase shift is implemented exactly, apart from a delay necessary to make the filter causal.

In sum, systems with generalized linear phase generally distort the phase (unless $\beta=0$ or $\beta=\pi$), but these distortions are desired when implementing differentiators and Hilbert transformers, and - apart from a delay - they implement the desired phase exactly.

As a final remark, note that the approximation of the response of an LTI system to a narrow-band input signal of bandwidth $W$ using phase delay and group delay is only valid if the magnitude of the system's frequency response is (approximately) constant in the band $[\omega_0-W,\omega_0+W]$.