Grant an app write access to external storage that does not request it? I would like to grant the Syncthing app write access to external storage in Marshmallow.  The app does not request it explicitly (unlike other apps that seem to use the Storage Access Framework to do so).  

In the app permissions screen, there is no permission listed for external storage, only for 'storage' (which seems to mean internal storage).  So there does not seem to be any direct way to grant this permission. 

My phone is rooted and I could just run Syncthing as root - but this option creates many problems and sometimes results in the app hanging indefinitely.  

Is there any other way around this problem?
You could manually add the related permission to write on external storage to AndroidManifest.xml but it won't work.

I found myself in a situation similar to yours, and I also tried to decompile the app, add the WRITE_EXTERNAL_STORAGE permission request to the AndroidManifest.xml and recompile the package, but it didn't work.

Since Lollipop, Android's new policy concerning the external SD requires any app which needs access to the above-mentioned to use the DocumentProvider interface. Unless you prefer to work in Smali, there's no way to implement it without the Java source code. Just editing the Manifest is not enough.

So, no. There's no way to manually add a working WRITE_EXTERNAL_STORAGE permission, neither in Lollipop or in Marshmallow. Your only choice is to ask the dev to implement it.