Just leave it running under System Account while you are troubleshooting, why complicate things?
If Logstash service fails to start and there is no log output it is usually because the JAVA HOME environment variable is not set (you can set this in the last "environment" tab on NSSM
As you say the service is definitely running, it can take 10 seconds before anything is written to the log.
Also, have you specified the log directory and level in the Logstash.yml file under logstash-x.x.x\config or have you specified this under the NSSM "Arguements" section. To keep things cleaner I prefer to specify the shipper file location and logging details in the Logstash.yml rather than using the NSSM arguments function.