Does relativistic quantum mechanics really violate causality?

The actual difference is in how these approaches treat measurements.

In the single particle theory, your observable is the particle coordinates $x^i(t)$. Measuring them at $t_1$ and $t_2$ can lead to apparent superluminal propagation.

In QFT, your observable is $phi(x) = phi(x^{i}, t)$. (I am ignoring the fact that these are operator-valued distributions). Measuring two of these separated by a space-like interval can not lead to superluminal propagation, as Peskin and Schroder show later when they evaluate the commutator of the fields. No grandfather paradoxes here.

OP has a point. On one hand, P&S on p. 14 argue that in first quantized RQM the propagator is

$$begin{align}&langle {bf x}_f,tau_f mid {bf x}_i,tau_iranglecr &~=~ int_{mathbb{R}^3} !frac{mathrm{d}^3{bf p}}{(2pihbar)^3} expleft[frac{i}{hbar}left( {bf p}cdot Delta {bf x} - Delta tau underbrace{sqrt{{bf p}^2+m^2}}_{text{Hamiltonian}}right)right]. end{align}tag{A} $$

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 flawed and not Lorentz covariant. A more careful Lorentz covariant analysis reveals that the RQM propagator is

$$begin{align}&langle {bf x}_f,tau_f mid {bf x}_i,tau_iranglecr &~=~ int_{mathbb{R}^3} !frac{mathrm{d}^3{bf p}}{(2pihbar)^3}color{red}{ frac{1}{2sqrt{{bf p}^2+m^2}} }expleft[frac{i}{hbar}left( {bf p}cdot Delta {bf x} - Delta tau underbrace{sqrt{{bf p}^2+m^2}}_{text{Hamiltonian}}right)right], end{align}tag{B} $$

cf. my Phys.SE answer here. Remarkably, P&S's above quote essentially still applies!

On the other hand, P&S in eq. (2.50) on p. 27 find exactly the same propagator (B) in second quantized real 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 energy states, so that one cannot dismiss negative energy states.

• There are unbounded negative energy states.

• The relativistic probability density $$ rho ~=~frac{ihbar}{2mc^2}left(psi^{ast} partial_t psi - psi partial_t psi^{ast}right) tag{C} $$ may be negative!

References:

1. M.E. Peskin & D.V. Schroeder, An Intro to QFT; p. 14 + p. 27.

The operator that measures whether a particle is at a particular position $x$ is the projection operator $mathcal{O}_x=|xranglelangle x|$. Suppose we have another projection operator $mathcal{O}_y=|yrangle langle y|$. In the Heisenberg picture, $mathcal{O}_y(t)=U^dagger(t)mathcal{O}_yU(t)$. Therefore, $$[mathcal{O}_x(t=0),mathcal{O}_y(t)]=|xranglelangle x|U^dagger|yrangle langle y|U-U^dagger|yrangle langle y|U|xrangle langle x|.$$ Now $langle y|U|xrangle$, and its complex conjugate $langle x|U^dagger|yrangle$, are non-zero, as shown in Peskin and Schroeder, even with the covariant integral discussed in Qmechanic's answer. Since $|xranglelangle y|U$ and $U^dagger|yranglelangle x|$ are not proportional to each other, the two terms cannot cancel. Therefore, the commutator is not zero, and a measurement performed at $x$ can effect a measurement performed outside of $x$'s light cone.