"It's probably not possible for the brain to both clean itself and at the same time [be] aware of the surroundings and talk and move and so on,"
So it would be the equivalent of a forced sleep, since you can't be conscious at the same time, but that might still be useful.
(Not sure precisely how different that is from the experience of most office workers ;) ).
Think of, for instance, the "language" areas (Broca's area and Wernicke's area). (The following over-simplified explanation assumes typical right-hand dominant individuals for the sake of brevity.) If you inactivate that area in the left hemisphere (using drugs or transcranial magnetic stimulation), you'll lose "language" processing of data that occurs around 1 Hz (in English, this corresponds to individual words, phonemes; in tonal languages, this includes tones of lexical significance). If you inactivate that area in the right hemisphere instead, you'll lose "language" processing of data that occurs an order of magnitude slower than that (in most languages, this corresponds to pitch and volume changes over the level of the sentence, and can convey prosody information such as speaker intent as well as speaker identity).
Due to specializations like this, hemispherectomies are not performed later in life. You would get symptoms similar to stroke patients. But for all areas, not just language.
In adults (or even adolescents) though, not as much luck.
In the case of hemispherectomies on Rasmussen's patients, you can analogize it as: people are dual-core by default. If you remove one of the cores early enough in life (like <5 yrs old), you can re-route the wiring so that processing can done by one of the cores, and still be pretty much normal. But whatever subtle thing is lost when if we don't do parallel processing like we usually do -- that things is pretty subtle to test for as well.
https://en.wikipedia.org/wiki/Cerebellar_hypoplasia#Signs_an...
There have been quite a bit of research on people with severed brain stems, where the brain halves effectively are cut off from each other and limited to communicating through external senses. In those situations the brain halves are actually exceedingly good at trying to obscure that anything is wrong by making inferences of actions carried out by one brain half that the other one could not possibly have knowledge of.
The bigger problem is that because the brain halves control different parts of the body, we'd likely lose a lot of our physical function.