Before I realized what the 'Analyze' link does, I did this on a board. It was a real pain in the ass setting and re-setting the board. I use it nearly every move.
Online Chess is meant to be like correspondance chess. Which, I believe, is mostly meant for studying. So, in this case, it is a great option. I find it has given me more confidence when I have to analyze in my head during live games.
As for ethical; I don't see an issue. I assume it is allowed, just like looking up opening lines, partly because it can't be prevented.
I think a lot of people don't use it. I see opponents make moves that seem like they spent about 2 seconds thinking about it. I can spend up to 30 minutes analyzing canidate moves.
Games that allow multiple days for movements, do you analyze them using the analysis mode that can be found on the same page as you make your move or on a real chess board during the game? Do you wait till after the game is over?
Is it ethical since it's offered to both players? How do you feel about analyzing ongoing games?