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.

$500 is the "I really don't want to do it" price.

I personally quote 5-10x the price of I don't want to do it.

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.

It probably just needs to be replaced.

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

It's just going to rot more. The rot needs to be cut out completely, and replaced with new (pressure treated) wood. Not just some putty and paint.

It's going to take most of a day to do that "right" because they will need to remove a huge chunk and replaced all the material. Don't believe these people saying a new door will be the same.... A decent exterior door costs $500 alone and install will cost at least that again.

That being said, I'd just replace the door. I recently replaced my front and garage doors myself. It's not as hard as you'd think. I spent like 4-6 hours on each. I did run into some leveling compound (cement type stuff) that leaked in from my flooring that I had to chip out on the front door.

Get a quote to replace the entire door and frame, it's probably cheaper.

Probably if the cost includes materials, his work looks good, and the labor is warrantied for at least a year. I'd probably opt to try to change out the wood on that frame in general though. You might want to see what your dealing with behind that wood, as it already looks like repairs have been attempted, and you want to make sure that this stops any issues back there more than superficially. If you go with the initial plan, make sure that the materials being used are of stronger quality/longer lifetime and rated for outdoor application. Get the warranty in writing somewhere if its not included on the invoice.