> Why wouldn't one expect the chaotic system to produce as much or more random output?
Because neurons as systems don't like random input. Noisy random input doesn't make (most) neurons fire. Rather the opposite, actually.
Neurons aren't (usually) simple threshold comparators of a single input; they're arbitrarily-complex gate-like recognizers, that fire in response to detecting signals in their combined inputs, or even upon matching temporal firing patterns within the signals of their combined inputs.
In that sense, if you tune every component in the system to produce more noise (chaos), you decrease the signal-to-noise ratio, and so the components that react to "important recognizable changes in the signal" within that noise have a harder time recognizing the thing they're looking for, and so they recognize it less, and so switch less.
---
By trite analogy: suppose you're a security guard. Your job is to recognize+report "real crimes in progress", while not reporting "false alarms." (Pretend that the police charge for false dispatches in your town, so your boss wants to optimize shrinkage vs. call budget.)
Also, your boss has decided to "keep you on your toes" by introducing "noise" into your computerized security system. Sometimes it'll show a burglary in progress when there isn't one. Sometimes it'll show that everything's fine, when actually the boss has secretly sent in a white-hat burglar of his own.
How willing to believe the security cameras should you be, to optimize your output signal to downstream demand?
As it turns out, you should probably try to figure out the false-positive vs. false-negative rate you're seeing reported by the downstream, and then calibrate your confidence up/down until you're making a balanced number of Type 1 vs. Type 2 errors. That level of confidence is the "critical point" — where the noise in the system is equally likely to help your detection (by "correcting" a false negative/positive, turning it into a true positive/negative) as it is to harm.
Moving away from that critical point, in either direction, makes your output less useful as a downstream signal.