Question on Wiener Filtering
On of the easiest ways to see why the outcome of the problem that Wiener Filter tries to solve is Linear Time Invariant (LTI) Filter is to derive (Solve) it.

By the very definition of the Wiener filter we're after a solution in the form of a filter applied using convolution which means this filter must be LTI.

Let's go through this.
The Wiener Deconvolution problem is given by:

$$ y \left( t \right) = \left( h \ast x \right) \left( t \right) + n \left( t \right) $$

Where:


$ x \left( t \right) $ is the signal to be estimated.
$ h \left( t \right) $ is the known impulse response of a linear time invariant system.
$ n \left( t \right) $ is some unknown additive noise, independent of $ x \left( t \right) $.
$ y \left( t \right) $ is our observed signal.


The Wiener deconvolution, in order to solve the above problem (Estimating $ x \left( t \right) $) is finding $ g \left( t \right) $ such that:

$$ \hat{x} \left( t \right) = \left( g \ast y \right) \left( t \right) $$

Minimizes:

$$ \mathbb{E} \left[ { \left( x \left( t \right) - \hat{x} \left( t \right) \right)}^{2} \right] $$

Since we defined the Winer Filter $ g \left( t \right) $ by the Convolution Operation we defined it as LTI.