Why Wiener filter not IIR
You're right, there usually is a better IIR filter (if you have enough data). The discrete-time Wiener filter is not "by definition" FIR. It is common to constrain the filter to the FIR case because it's often more straightforward to implement, and because such a filter can be made adaptive more easily. Also, in practice you often want to consider only a finite data window, which makes the optimum Wiener filter FIR.

However, you don't need to impose the constraint that the filter length be finite. The most general causal discrete-time Wiener filter for data available over the infinite past is an IIR filter.

A derivation of the FIR and the IIR cases can be found (among others) in Digital Signal Processing by Proakis and Manolakis.

EDIT: If we only have a finite amount of data available, the optimum Wiener filter has a finite impulse response. Assume we have $N$ data points (one current and $N-1$ consecutive past data points), then the best we can do with a linear time-invariant filter is to linearly combine those $N$ data points, by multiplying each data point by a coefficient and adding them up:

$$y[n]=\sum_{k=0}^{N-1}h[k]x[n-k]\tag{1}$$

The coefficients $h[k]$ are optimized such that the mean square error between the filter output $y[n]$ and some desired signal is minimized. The $N$ coefficients $h[k]$ are the impulse response of the corresponding FIR Wiener filter.