Rotating two chains on one cassette I clean my chain on the bicycle, mostly because taking it off for a bath, drying, etc seems to take more time than I have. But maybe I could use 2 chains in rotation, I would take one chain off, replace it with second clean one, and during the days that follow would slowly clean it to be ready for next cycle.

Are there any drawbacks to such approach regarding the wear of cogs, or something else?
There is actually a real gain from this strategy. By rotating 2 (or 3) chains you extend the life of rear cogs (cassette) and it will last all your chains. In case you use one chain, you may need to replace cassette together with chain.

As of front cogs, they last longer, but you will get a more even wear.

Speaking of drawbacks - regularly taking off your chain you can make your PowerLink (or other brand) a bit more loose and have a higher risk of it failing under a load.