The way that I and most of the players I have observed do a capture is to pick up the capturing piece with one hand, swoop in and remove the captured piece with the same hand, and place the capturing piece on the board in one continuous motion. I have rarely seen a player remove the captured piece first.
If there is a simultaneous capture and promotion with a pawn, a player may pick up the Queen with the other hand to place that piece on the board to complete the capture and promotion process. I generally make the capture on the 8th rank and then get a Queen to replace the pawn. After that I press the clock.
Most people seem to do it the way you describe. I sometimes pick up the captured piece first and with the same hand then grab an move the capturing piece.
There is no rule on which you should do first. It’s your option. The thing to keep in mind is that the touch move rule will apply first to whichever piece you touch first. If you pick up your own piece to initiate the capture, and then before touching your opponent’s piece decide not to make your intended capture you will still have to move your piece, but will not have to capture the opponent’s piece. If you initiate the capturing sequence by first picking up your opponent’s piece you will be required to capture that piece if you can legally do so.
While working the recent Pan American Intercollegiate I noticed that most of the top players where capturing by taking the opponents piece off the board first.
I noticed a ‘two hand’ process in speed games among the world’s top players - take off the captured piece with one hand, replace it with the capturing piece in the other. Perhaps they think this is faster to execute.
If those are casual games, fine. If those are tournament games, then the practice of using two hands to capture is disallowed under both the FIDE Laws of Chess and the US Chess Official Rules of Chess.
If you are castling by playing Ke1-g1 and Rh1-f1, and you are simultaneously capturing two of your opponents pieces en passant at f1 and g1, in what order do you touch the four pieces?
Gotta love gratuitous rules differences. Of course, we wouldn’t want to burden US tournament players with one consistent set of rules everywhere, now would we?
The practical advantage of this…despite any engineering objections based on arm motion…is that the rules dictate that the touched piece must then be captured. That typically drastically reduces or eliminates the number of blunders that could occur (e.g. grabbing a bishop to capture a pawn only to find out you grabbed the wrong square color of bishop!?).
I suppose, although isn’t it at least as plausible that you reach out and grab the wrong piece? Maybe it depends on how your brain sorts this stuff out: is the focus on the captured or the captor?
Always plausible, just an observation that there are typically less capturing possibilities than there are move possibilities.
A secondary influence is equipment. No one wants to be “that guy/gal” who knocks over pieces on a DGT board. The prevailing “drop down from the sky” action to pick up the captured piece seems more deliberate than “swoop and swap.”