It also dawned on me your method is basically the same as vonNeumann's 1946 middle square method, which turned out to be terrible from short cycles. His was base 10; yours is base 2. His dropped the bottom and top "bits". Yours drops the top (during the mult). His permuted each time via a different number; yours uses a fixed number.

Both use a mult to scramble, and then retain a subset of the bits. Adding more rotates and mults does not change the underlying issues.

This type of PRNG has been pretty thoroughly mined over the past 70+ years. You should dig through this work and the following literature on why PRNGs of this structure fell out of favor. I suspect exactly yours appears somewhere following this work.

https://en.wikipedia.org/wiki/Middle-square_method