I know nothing about antivirus software, but isn't security software supposed to be open? Otherwise, it's just security through obscurity. It sounds to me like Symantec just wants to hide all their vulnerabilities.
As you point out, it's security through obscurity (which we all know isn't really security), but what's more, you can never know if there is a backdoor in proprietary software, or if it's actually doing what it's supposed to do (and not less, or more - like logging your activity or listening in on your traffic).
How would you even know, unless you saw the source code?
So while you probably wouldn't know via technical means, my gut feeling is that a conspiracy in a company that large would quickly surface.
This is antivirus software, a specific sub genre of security software. Historically, the most popular such packages have not been open.
You are not expected to trust them not to have backdoors any more that you are expected to trust any other vendor (say, Microsoft). What you are expected to do is trust that it catches virus and third party spyware. Which, supposedly, it does, and people have been using it for ages.
Even if it was open, you would need to have it in binary form to run it, so you either need to know to compile it yourself and check the code first (not an option for 99.9999999 of the users) or trust the party that compiled it for you. And then you need to check again for every virus definition and engine update downloaded.
Better just trust the vendor and use it closed source...