You can get a new door and frame installed for about the same cost. Once rot sets in, it has to be cut out completely which then further compromises the frame. Putty, caulk, and paint will just slow down the inevitable.

If you want it done right I'd go ahead and find another handyman who can hang a new door. It's not a terribly difficult job but steps have to be taken to keep water from ruining the new door frame, it needs to be leveled, etc.

EDIT to add - is this a single door or a French door? I can't quite tell by the pictures you shared. If it is indeed a French door then the cost will be quite a bit higher.