Why do Pentagon tilings not solve the Einstein Problem? The einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles, that is, a shape that can tessellate space, but only in a nonperiodic way. (This Numberphile video has more context).
Now there are pentagonal shapes that tile the plane in a non-periodic way.

Why are these shapes no solution to the einstein problem?

I suspect that I misunderstand some criteria of Aperiodic set of prototiles, but i cannot figure out what it is.
 A: They key is that such a tile can only tile the plane aperiodically.  The other tiles you cite do not satisfy this criterion because while they can tile the plane in an aperiodic fashion, there exist other tilings of that shape that are periodic.
A: To add to heropup's answer, notice that even a square can tile the plane non-periodically (assuming by non-periodic we need a full rank sublattice of translational symmetries); just shift a subsequence of the columns in the usual integer lattice displaced tiling by some non-integer amount.
Take on the other hand the set of Robinson tiles. The shapes are important here, the decoration with lines on their interior just helps to prove their aperiodicity:

These tiles can tile the plane but the only way they can is in a non-periodic fashion. It is impossible for these tiles to tile periodically. These two criteria are necessary to be satisfied in order to be called an 'aperiodic' set of prototiles.
The full tiling looks as follows, where only the decoration has been left intact for the majority of the image:

