Thanks for your input everyone! There are two separate issues at play here. One is the option to play full capture sequences at once, and the other is the way the moves/captures are executed on the board (automatic, one-click, etc).
We have put this first issue on the roadmap, and are planning to add a "full captures" setting for the analysis boards (which would apply to analysis, studies and puzzles when enabled). Live games are still a different matter for reasons outlined earlier, as your way of moving the pieces there does not only affect your own play, but also that of your opponent.
With this new setting enabled you could drag or click a piece immediately to the ending square of a capture sequence, without having to perform all the steps in between yourself. When there are different capture paths possible, you would first have to click to a point where the paths diverge, after which you can click to the ending square.
A different issue is the way moves can be executed on the board, for which several proposals have been made as well. We think fully automatic captures can be safely discarded for the reasons @Tim outlines (it will make things messy and confusing). For the same reasons, it doesn't seem like a good idea to move pieces by clicking a target square alone, in cases where there is only one piece that can move there. The most intuitive way to move a piece starts with clicking/dragging the piece you want to move. Mixing in other ways (that only happen in certain situations) hardly adds any value, but potentially brings a lot of confusion.
Single-click moves are a different matter (clicking a piece with only one move will automatically play that move, capture or not). This is still initiated with a click on the piece you want to move, and thus is most natural. On the other hand this could also be confusing for people that aren't used to it, because you have to keep in mind that the board behaves differently in some situations. For example, many people show the habit of hovering pieces while thinking, deciding whether to drop it or play something else. It is especially annoying with puzzles when this results in an unwanted move being played immediately.
We would love to hear more opinions from the community about this. Our preference is to only move pieces by drag-and-drop, or by clicking the piece followed by its target square, regardless whether the move is forced or not. There is certainly value in keeping the way of moving pieces consistent across all boards and positions. However, if the consensus is that one-click moves are an addition as well, this can be included as a side-effect of the new "full captures" setting (in which case something like "quick moves" might be a more appropriate name).