I would imagine that it won't generate enough power for a Wii controller. At the very least it would be bad user experience if for some reason your ability to interact with the game occasionally dropped out because you weren't in a 'vigorous' enough part of the game.
The wii controller is constantly using (relatively) large amounts energy to measure/process/send sensor data for however long you play. A remote on the other hand is intermittently transmitting short messages. My guess is that a large amount of the energy dissipated in the remote batteries is self-discharge, which makes putting a normal battery in one a little bit of a waste.
Looking at the power consumption of the various components in the Wiimote, the only thing that really eats a lot of energy is the radio. Data is not processed on the Wiimote. Looking at the data stream coming out, the only processed data is the camera at the front, which does it super-efficiently in an ASIC/DSP. Maybe a supercapacitor or normal battery in addition to one of those could store energy from the vigorous parts and allow for unbroken interaction. Games are already designed to pause on battery low, so it would not be critical, just annoying.
I know there's not serious data processing on the remote, but I count an analog to digital converter as processing.
The main reason why I don't think it would be suitable for a wii remote is that two AA batteries are supposed to last 45-60 hours depending on use. Remotes last much longer. Granted the wii remote is used more actively, I still think the non-reliability issue, even if only for pause, kills it as a power source.
In the end though, its probably the consumer's decision unless nintendo hard wires them in.
The Audio comes in as ADPCM, which does need to be decompressed and then goes through a DAC plus presumably some kind of amplification. ADPCM is easy to decode, but does still need a processor of some kind.
The proximity chargers for Wiimotes work really well. Don't even have to take off the outer rubber sleeve to use them. I'm all for reducing use of standard batteries, so at least the Wii still has a decent solution for this.