A proof of concept already exists
This is not a thought experiment. Caitlin Long's Custodia Bank in Wyoming was built from the ground up as a 100%-reserved, non-lending institution — it holds the full value of fiat deposits in reserve and does not engage in fractional reserve banking at all. Wyoming's special-purpose depository charter, and similar statutes now in a handful of states, exist specifically to allow this model. Custodia has spent years in court fighting for a Federal Reserve master account precisely because the incumbent system does not want to let the honest version of a bank plug into the rails on equal footing.
That fight tells you everything. A fully-reserved bank is not a fantasy. It is operating. The resistance it meets is the resistance of a model defending its franchise.