The way you put it, it seems like you want to care for both X and Y but are unable to.
“I want to care for an arbitrarily chosen person from the set of X and Y” is not “I want to care for X and Y”. It’s “I want to care for X or Y”.
“I want to care for an arbitrarily chosen person from the set of X and Y” is not “I want to care for X and Y”. It’s “I want to care for X or Y”.