Maybe I can't see it in your pics, but I don't see a fallen gutter, just bad facia and fallen downspout parts.
I have only worked on one house with wooden gutters and the repirs we made were fashioning one end that was bad and repairing lead linings on previously repaired ones. My feeling (from what I get from your pics) is that there has been a bad drainage issue previously and water has gotten behind the gutters and rotted the facia. The gutters will have to come off the replace the facia and I can't tell the condition of the gutters but if they are in bad shape, they might not last that operation.
If you can afford to replace all gutters and facia it certainly would do your house a big favor, especially if you use Azek or similar facia, as it will never rot. But, that said, it would behoove you to find out what caused the leaking/damage and make sure the new installation would not be prone to the same issues.
Whether you think you can, or you think you can't, you are right. - - - Henry Ford