No: complex-valued continuous functions are Baire measurable. THeThe real and imaginary parts of a $\mathbb C$-valued continuous function are $\mathbb R$-valued continuous functions.
Note: even for compact $X$, you do not use merely $G_\delta$ sets to generate the Baire sets, but closed $G_\delta$ sets.
When the compact space $X$ is a metrizable compact, Borel=Baire and all measures are regular. When $X$ is not metrizable, it could happen that two different Borel measures induce the same linear functional on $C(X)$. So, to represent the dual space $C(X)^*$, you can either (i) use Baire measures, or (ii) require regularity.
A reference:
Varadarajan, V. S., Measures on topological spaces, Am. Math. Soc., Transl., II. Ser. 48, 161-228 (1965); translation from Mat. Sb., n. Ser. 55(97), 35-100 (1961). ZBL0152.04202.
Of course it considers not only compact $X$. For completely regular Hausdorff $X$ we can define the Baire sets as the sigma-algebra generated by the zero-sets. [Here, "zero-sets" are sets of the form $\{x \in X : f(x) = 0\}$ where $f : X \to \mathbb R$ is continuous.] If $X$ is compact Hausdorff, these are the same Baire sets. But in general $$ \{\text{compact }G_\delta\} \subsetneq \{\text{zero-sets}\} \subsetneq \{\text{closed }G_\delta\} $$