Too much hussle for gains that are inferior to simply killing everything. We already have sort of "different mission goals". The player sets them.
In some missions like cult outpost it is capturing head honcho, you don't have to do anything else, if he spawns just next to your craft, you can stun him, grab his body, take it back and abort mission. Here you go, you won the mission, "case" might not be closed, but you gained something, a "material" to research, to progress on tech tree.
"Steal Something"? Whole point of Cult HQ is to get that specific thing (red card, gold statue) and after getting it, Agents can just fly away and X-Com can call this mission a success.
Apparently what you want was always implemented, even in vanilla X-Com, you just have to look outside the box.