This is something that has stumped me, and I am sure there is a simple answer that I am not finding.
When I cut a path, and then have a bunch of unjoined paths, and want them to be joined to become one shape, when I chose Path, Join, it connects the dots, but the logic doesn’t seem logical to me.
AI is guessing based on some magic Adobe math. Sometimes it’s good, sometimes it’s really dumb.
To force it to do what you want - use the white direct selection arrow or lasso to select two overlapping points on different paths at a time and join them (ctrl/cmd+j), then select the other two and join them. Continue until everything is done.
Ps) If there are a lot of segments to join, and they are all touching you can cheat and use the bucket to make a live paint object, expand it to get a filled shape, then just turn off the fill. Depending on how the object is built, you might have a few extraneous lines to delete.