No it cannot run on fat.

Strong statement. The brain can run on ketones just fine, it's constantly titrating energy consumption from different sources. You might say the brain uses glucose first as a priority, but that is also true for alcohol.

Fat can be broken down into glucose, though

Not really a complete statement, gluconeogenesis requires more inputs then just fat, so of the glucose produced only 15% of the energy is coming from fat.

There is a 1972 paper that demonstrates normal functioning after prolonged fat adaptation, it's a great read, but this specifically speaks to brain glucose utilization - https://doi.org/10.1172/JCI107095