She does live in Monaco. Just not at the palace. They also have a family home a stone throw away from Monaco.
Her brother and his family lives in Monaco. Presumably, if this was planned treatment she was undergoing, she would have done so closer to home.
Her brother Gareth lives there. Pre-covid her family visited regularly.
I do think people are reading too much into these tears. After the ceremony at the palace, they went to Sainte Devote church so that she could leave her wedding bouquet at the altar (Sainte Devote is the patron Saint of Monaco). There was a gathering of a selected few, and she started crying when a religious song was sung by a parishioner. Isn’t it normal to feel emotional during one’s wedding, specially when things are being played out in front of the world. She wasn’t the only one crying : people were very moved in that church, one of the reasons being that it was a beautiful moment ;-)