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.
Rot doesn’t necessarily need to be cut out. There are epoxy products that will wick into the fibers and strengthen them, and then epoxy putties that can be used over the top prior to painting, which will also have the side benefit of being more resistant to rot than wood. I’m not sure that I would use those products in this case though; replacing the frame would probably be less of a hassle.
It probably just needs to be replaced.
It’s probably a fair price given the amount of manual labor involved. I didn’t think it would be much until I did the job on my own French doors earlier this year.
Watch a few YouTube videos on replacing these pieces. Maybe it will build some confidence in trying it yourself. It isn’t a complicated repair and I’ve stopped calling contractors to do small jobs anymore. In the last year I’ve permanently wired in a backup generator to my house and installed a 220 receptacle for a new induction stove. I’ve always been afraid of electrical work but it was so easy and saved me $1000
