My understanding is that neurons are like a higher-order connected mesh. A bit like many fully-connected layers in a Machine Learning model.

You can't just read fine motor signals from any neuron, but given enough neurons, since they're deeply connected, you can infer a close approximation of the intended motor signals given a known output. That's why they need a lot of sampling wires in the brain; to makes sure they have enough data to make that inference. Luckily, machine-learning is very good at figuring out patterns in this sort of data.

My guess is that they train an ML model using the inputs and outputs when the joystick is connected. Then they can use the trained model to just generate the output directly from the inputs when the joystick is disconnected.

This will work fine for one particular monkeybrain/interface combo; however, you cannot use the same model on other monkeys. You have to train a new one each time.