I've yet to see a sliding or french door installed the right way, or have it properly maintained.
There should have been a sill pan under the doors threshold.
Jamsill Guard Door & Window Sill Pan Flashing
Within 24 hours that door frame and door should have been painted.
There should have been a piece of vinyl lumber installed under that threshold to stop it from flaxing on the outside wall tight up againt the bottom side of the threshold.
Any new doors we have been ordering have soild vinyl jams, and vinyl brick moulding. Yes they will cost more but only having to buy one door for the rest of your life will make up for it.
Most of the time when a door fails and there's inside subflooring damage it's because the stoop, deck, porch was built so it's even with the door opening.
A guarrentyed way to have it fail. It should have been at least 4" below the threshold. A standard step is 7" so that would have been the best height.
There is no good way to just patch up your old door and not have it show. By now not only is the brick moulding and jams rotted there's damage to the subflooring under the door.
No form of caulking or wood filler is going to fix the real cause of the problum and waiting to see if it works is just going to end up costing more money and time.
Just step on the threshold and twist your foot, look in the corners of the threshold and you will see the threshold flex and the seam in the outside corners will open up.