@xyzzy because that's the much easier solution, just look at how unproblematic they solved that pope problem with the slave contract as background. And the same goes for all the other stories, it has first the "historic" background of being common in that time and age, and most MC's are intelligent enough to not just go completely against the standards and rules of that time, while remaining the mentality of his former life, that is, treating a slave equal to himself.
"MC could have just said that she could stay with his party for as long as she needs to without having to physically own her." yea, he could. But since the physically own part never crosses his mind, it ain't a big thing for him. He says himself that one more slave is rather a burden than having one more free worker. A slave in isekai standards, in MC's perspective, usually equals family standard, that is the rule so far. The wording Slave makes things just easier. Even funnier when the slaves want to remain slaves... pls stop taking isekai "slave" word using serious.