Correlation function and spontaneous symmetry breaking

Let’s look at ferromagnetic system as an example.

The correlation function is defined:
$$G(r,r_{0})=langle(m_{r_{0}}-langle m_{r_{0}}rangle)(m_{r}-langle m_{r}rangle)rangle=frac{1}{r^{p}}e^{-frac{|r-r_{0}|}{xi}}$$.
In my understanding, When $G(r,r’)neq 0$, it means the system is somewhat ordered and $G(r,r’)=0$ for disordored system.

When study the impact of dimension on whether spontaneous symmetry happens, we may consider the tranverse correlation function
$$G_{perp}(r)=int text{d}^{d}k G_{perp}(k)e^{ikr}propto int text{d}k k^{d-1}frac{e^{ikr}}{k^{2}}. $$

The textbook I’ve read says:

When $dleq2$, in the limit $krightarrow0$, $G_{perp}(r)$ is divergent and hence the ordered state will collapse thus no spontaneous symmetry happens.

I understand that since the susceptibility function, which measure the influence of change of external field in position $r_{0}$ on the order parameter in position $r$:
$$chi_{r,r0}=frac{partial m_{r}}{partial h_{r_{0}}}=frac{1}{k_{B}T}G_{r,r_{0}}$$
is also divergent. So a infinisimal fluctutaion of $h_{r_{0}}$ will cause amountable change in $m_{_{r}}$

Here is the problem, divergece of correlation function seems to keep the system in a disordered system. Yet in the contents in bold font, I think non-zero correlation function means order. These two clearly contradict. Could anyone help to give a clearer physical picture?

Another question maybe relevant to this is: spontaneous symmetry always happens from state with higher sysmmetries to state with lower symmetries. How come a disordered system can have more symmetries than an ordered system?