Why does a MOSFET have 2 semiconductor leads?
What you are proposing is called a Schottky-Barrier Source and Drain MOSFET:
Compare this n/p well MOSFET

with this metal source/drain MOSFET

Images from https://www.semanticscholar.org/paper/Electrical-Characterisation-and-Modelling-of-metal-Pearman/f3d7a94a98b10c7c36e527c697eb3d158d69d4c0
See also https://people.eecs.berkeley.edu/~tking/theses/rvega.pdf
The metal S/D lowers the Rds and improves the ft, but the Schottky barrier doubles the threshold voltage.