We use an own users for security reasons.
You could run it as any user like nobody or your normal user account as well, it’s “just” state of the art.
Just beware of running it as root or user that might that sudo applications without any authentication. This is a potential security issue then.
I didn’t dig too deep into Heroku to be honest, as the Feature page is quite big and it gets too much at some point (kinda out of scope here). Personally I don’t think you should do this, because you’d be missing your elasticsearch service. Also, depending on how fast your database in Zammad is growing, that service might get expensive quite fast.
You could take a look at the following virtual servers, I use a VPS S SSD for 5 Bugs a month. Runs totally fine.
Don’t get me wrong, but I personally would prefer my own administration over something semi administred. Oh and by the way: If you don’t want to care about hosting stuff, you could also use our hosted services. 
Edit: Forgot the contabo link: https://contabo.com/?show=vps