Bose-Einstein condensation is not a quantum phase transition.
Quantum phase transitions occur only at zero temperature, and become crossovers for any $T\neq 0$. In the same way that the paramagnetic-ferromagnetic Ising transition only occurs at zero external field, and becomes a crossover for any $H\neq 0$.
Experimentally, of course, you'd only probe the $T\rightarrow 0$ limit but you'd still call it a phase transition. Because, why not. So you'd (practically) consider quantum phase transitions happening also at $T\neq 0$.
The reasoning behind all this is that there are two energy scales$^\dagger$:
- thermal: $k_{\mathrm{B}} T$
- quantum: $\hbar \omega$
Only when $T=0$ the thermal energy scale vanishes completely and the quantum one can dominate and control the dynamics. In the Ising analogy, these correspond to the external field interaction energy $\mu H$ and the inter-particle interaction $J$ respectively.
Hence, in quantum phase transitions, temperature drops out completely. The transition is then driven by something else. An order parameter. Magnetisation, disorder strength. Interaction strength. Both. Etc.
Classical (temperature-driven) phase transitions are characterised by power-law divergence of several parameters, for example the correlation length $\xi_\ell$ and the correlation time $\xi_t$:
$$ \xi_\ell \propto |T-T_{\mathrm{c}}|^{-\nu}, $$
$$ \xi_t \propto |T-T_{\mathrm{c}}|^{-\nu'}. $$
We could come up with a characteristic frequency scale by taking $\omega_c \sim 1/\xi_t$:
$$ \omega_c \sim 1/\xi_t \propto |T-T_{\mathrm{c}}|^{\nu'} \rightarrow 0 \quad \mathrm{as} \quad T\rightarrow T_{\mathrm{c}}.$$
The above argument is sometimes used to justify why the quantum energy scale $\hbar \omega_c$ is always going to be shadowed by the thermal one in the vicinity of $T_{\mathrm{c}}$, thereby not playing any effect. And requiring $T=0$ to really be dominant.
Contrary, in Subir Sachdev's book I find a graph (Fig 16.4) that
suggests BEC is a quantum phase transition like any other, with a
quantum critical point at $T=0$ and $\mu<0$. As a specific example, we could
consider a weakly-interacting Bose-gas in three dimensions.
Bose-Einstein condensation is a non-interacting effect. The fact that a macroscopically occupied ground state occurs despite interactions is more complicated and, I would say, another matter. As you said, BEC is driven by particle statistics only.
I don't have the book but I am assuming the figure you are referring to is the the same as fig. 3 here. In eq. 50 on the same page he has a dependence $\mu/u_0$ where $u_0$ depends on the interaction strength $a$. So in the non-interacting limit $a\rightarrow 0$, that goes away and the reasoning stops being applicable.
The chemical potential cannot be thought as the external parameter driving a quantum phase transition as you don't have independent control over it. It is determined by the system itself, as explained here.
$^\dagger$: Though I am not aware of situations where these two energy scales directly compete against each other, there are phase diagrams (e.g. unitary Bose gas) where you can go from BEC to thermal Bose gas by varying $T$, and from BEC to unitary BEC changing the interaction strength $a$. These are all crossovers but still, food for thoughts.