OP has a point. On one hand, P&S on p. 14 argue that in first quantized RQM in the Newton-Wigner (NW) operator formalism the kernel is
$$\begin{align}&{}_{\rm NW}\langle {\bf x}_f,\tau_f \mid {\bf x}_i,\tau_i\rangle_{\rm NW}\cr
&~=~ \int_{\mathbb{R}^3} \!\frac{\mathrm{d}^3{\bf p}}{(2\pi\hbar)^3} \exp\left[\frac{i}{\hbar}\left( {\bf p}\cdot \Delta {\bf x} - \Delta \tau \underbrace{\sqrt{{\bf p}^2+m^2}}_{\text{Hamiltonian}}\right)\right]. \end{align}\tag{A}
$$
P&S write on p. 14:
This integral can be evaluated explicitly in terms of Bessel functions. [...] the propagation amplitude is small but nonzero outside the light-cone, and causality is violated.
See also this Phys.SE answer. However, P&S's normalization of the integrand (A) is not Lorentz covariant, and hence not suitable for RQM. A more careful Lorentz covariant analysis of the path integral formalism reveals that the RQM kernel is$^1$
$$\begin{align}&\langle {\bf x}_f,\tau_f \mid {\bf x}_i,\tau_i\rangle\cr
&~=~ \int_{\mathbb{R}^3} \!\frac{\mathrm{d}^3{\bf p}}{(2\pi\hbar)^3}\color{red}{ \frac{\hbar}{2\sqrt{{\bf p}^2+m^2}} }\exp\left[\frac{i}{\hbar}\left( {\bf p}\cdot \Delta {\bf x} - \Delta \tau \underbrace{\sqrt{{\bf p}^2+m^2}}_{\text{Hamiltonian}}\right)\right], \end{align}\tag{B}
$$
cf. my Phys.SE answer here. Remarkably, P&S's above quote essentially still applies! There is a superluminal overlap.
On the other hand, P&S in eq. (2.50) on p. 27 find exactly the same kernel (B) in second quantized KG QFT. So OP is correct that RQM appears in the one-particle sector of free scalar QFT.
P&S write on p. 28:
So again we find that outside the light-cone, the propagator amplitude is exponentially vanishing but nonzero. To really discuss causality, however, we should ask not whether particles can propagate over spacelike intervals, but whether a measurement performed at one point can affect measurement at another point whose separation from the first is spacelike.
And P&S then go on to show that the commutator $[\phi(x),\phi(y)]=0$ vanishes outside the lightcone, so that real KG QFT is causal.
One problem for first quantized RQM (which OP seems well aware of) is that it does not describe particle creation and annihilation per se.
Also the usual objections to first quantized RQM still apply, such as, e.g.:
Local interactions couple to both negative & positive frequency states, so that one cannot dismiss negative frequency states.
There are unbounded negative frequency states.
The relativistic probability density
$$ \rho ~=~\frac{i\hbar}{2mc^2}\left(\psi^{\ast} \partial_t \psi - \psi \partial_t \psi^{\ast}\right) \tag{C} $$
may be negative!
See also this & this related Phys.SE posts.
References:
- M.E. Peskin & D.V. Schroeder, An Intro to QFT; p. 14 + p. 27.
--
$^1$ Both kernels (A) and (B) satisfy the kernel property
$$ (\Box_f-m^2)\langle {\bf x}_f,\tau_f \mid {\bf x}_i,\tau_i\rangle~=~0, \qquad \Delta\tau~:=~\tau_f-\tau_i~>~0,\tag{D}$$
but only propagators (A) satisfies the normalization
$${}_{\rm NW}\langle {\bf x}_f,\tau_f \mid {\bf x}_i,\tau_i\rangle_{\rm NW} ~\longrightarrow~\delta^3(\Delta {\bf x}) \quad \text{for} \quad \Delta\tau \to 0^+. \tag{E} $$
