You may want someone at Alteryx to confirm but part of the confusion working with Gallery is that in most cases it will be running under a service account...i.e. not your credentials.
It also packages flows and such to store in a MongoDB which means you can't do some things like dynamically alter a flow you will chain to. Its default behavior is to package macros which is a maintenance issue if you have lots of flows sharing macros and you change a macro...then you have to manually open and republish every flow using it, there is a way around the macro issue but that is a different thread. You also can't chain to a flow external to the Gallery/Studio. It runs in a sandbox of sorts and under credentials other than yours, unless you specifically define the "Run-As" option.
As such I believe when you save it does some validation using the credentials that will run the flow in Gallery access the input and output paths, again someone at Alteryx can confirm.
From what you say and show I am thinking the credentials associated with the Server/Gallery does not have access to write to that output path. If you did not specify to Run-as "you" it will run under the Server credentials and if it says "cannot access folder" then it would seem the Server credentials don't have write access, since it is associated with an output tool, to the folder.
I am pretty new to the gallery myself and it is an adjustment. Just because it works when you run it locally to test things doesn't mean it will work in the gallery.
Just keep in mind the credentials typically will change so you may need to talk to your Admin to get access rights for the server credentials modified. Also the environment changes..things aren't necessarily where you think because of packaging and storing it in a DB.
