Syncthing logs to standard out. What happens to that standard output depends on how it’s being run. Are you running it from the command-line? Launchd? etc? Whatever mechanism you are using to run it, check what that mechanism does with its stdout.
I found a solution to debug. I just started the application over the command prompt and the error message was shown in the terminal. It was a permission error. Now, everything is working fine
Now, I use Homebrew too (awesome) but for something like SyncThing, it really doesn’t make much sense to use it in my opinion. Homebrew is great for things with lots of dependencies/options, say php, but SyncThing being a single binary it seems overkill. Just my opinion.