There are often multiple good replies for defending a tactical position and often a particular defense might be presented to train a particular pattern.
Any system that had the solver doing both sides would require the system to allow multiple solutions and often no clear resolution other than a winning (or drawing) evaluation.
It could be possible to sort out those puzzles that have only 1 decent move for both sides, and create a puzzle system with 2 player mode or 1 player playing 2 sides. That would however be a slightly different mini game than what is now implemented here. The idea is great, and maybe some day there will be such a game variation. It only takes some coding and smart algorithm to sort out such puzzles. The scale of puzzle types would be slightly limited, but the number of possibilities is still high.
I assume this has been talked about at some point but I couldn't a discussion on this and there's probably a reason why this is not done this way, but if not, I figure it's a reasonable discussion.
There's two ways to solve chess puzzles under the current mechanics, solve it prior to the first move, or finding a good first move and solving reactively to subsequent moves. A lot of puzzles have their ratings skewed bc under certain conditions, the difficulty of a given puzzle for those that solve reactively and those that solve the puzzle before moving can be monumental. Having the user move both black and white solves this by only having one way of solving puzzles.
For example, this puzzle here: https://www.chess.com/puzzles/problem/558470. There's only one obvious safe check, then black sacrifices their queen to prevent a M8, where you just finish the puzzle by taking the sacrificed queen. For someone at this rating who solves the puzzle prior to moving, you have to calculate out the full M8 forced sequence otherwise the check just looks like a check for check's sake. For the reactive solvers, that's the only safe check so you go for it and then after the queen sacrifice, it is obvious you just take the free queen.
I'm not trying to say there's a right or wrong way to solve a puzzle, I'm just saying that many puzzles create conditions where the two primary ways to solve the puzzles result in drastically different difficulty levels and skewing the score rating. Requiring the user to move both sides would result in significantly more consistent ratings and difficulty for puzzles for all users since there's only one way to solve the puzzle.