Alright will probably change it out. I'm assuming I could pick up a door frame from home Depot for $50-100.
How hard to change it out?
Do they come a standard door hight size or will I need to cut it shorter?
Just peel the trim off and pull a few nails out and it will come off?
I like your original idea of cutting out the bad section and patching it up. I would take a putty knife and work it under the door stop to break the paint seal and then pry it off being careful not to break it. Then do the same with the door trim and remove that. The door stop is away from the floor as there was a threshold strip across there filling in the gap that has the plastic tape on it now. You can buy a new one of these at the builder center. I would then mark a line high enough to get above the damage and saw cut as far I could get across the bad part and finish off the cut with a chisel. Patch in a piece and I like to use automotive “bondo” in the cracks and sand out just like I was patching a car body. Paint and replace trim.
If you start prying off that old door jamb with my luck I would end up hanging a new door by the time I was done. If everything else looks good enough just fix the bad spot IMO.
If it doesn’t work you can always go to plan B. you will just be out a little time.