Both Chet Miller and Buck Thorn have provided good answers, and explained that
$$G = \sum \mu_i N_i \tag{1}\label{1}$$
is indeed a function of $T$ and $P$. One way to show that is as follows.
As you say, we can derive (1) from the Fundamental Theorem of Thermodynamics for a closed system in which composition can change:
$$ dG = V dP -S dT + \sum_i \mu_i dN_i \tag{2}\label{2}$$
Now take the differential of (1):
$$dG = \sum \mu_i d N_i + \sum N_i d\mu_i $$
Equate that differential with the Fundamental Theorem (2), to give
$$\begin{align}
dG = \sum \mu_i d N_i + \sum N_i d\mu_i &= V dP -S dT + \sum_i \mu_i dN_i \\
\sum N_i d\mu_i &= V dP -S dT
\end{align}$$
which shows that a change in either $T$ or $P$ at constant composition necessarily results in a change in some chemical potential $\mu_i$. Hence, $\mu_i$ is a function of both $T$ and $P$, as is $G$ in equation (1).
For a system composed of a pure substance,
$$\begin{align}
N d\mu &= V dP -S dT \\
d\mu &= \overline{V} dP -\overline{S} dT
\end{align}$$