I don't know where you got your info from, but some of it is correct, some of it is incorrect.
Does rooting open up security vulnerabilities? Technically rooting generally exploits security vulnerabilities in order to gain root access. The security vulnerability was there regardless if you're rooted or not. Security holes are generally patched because the phone manufacturer/carrier probably check out forums for root exploits in order to patch them. However, on the Galaxy S3, the bootloader is unlocked (except the VZW version that is locked for the meantime) and no security exploit is necessary to gain root.
So how does rooting open up security holes? Well, the only way it really does is through the SuperUser app's (or SuperSU app) vulnerabilities. Which is constantly updated to remove such vulnerabilities. Plus you could flash a ROM that adds vulnerabilities, but the devs have to put their source out in order to comply with this thing called a GNU. Developer websites will not allow a ROM developer to host their files without providing a source, so that the security can be verified.
Rooted phones can actually be more secure. This is because rooted phones can get updates much more quickly than non-rooted phones that need carrier-initiated updates. For example, you may have heard of Carrier IQ? My previous phone, the Evo 3D, had that installed. Because I was a rooted user, I was able to remove it on the spot or flash a mod that removed it within a day of it being discovered (by an Android ROM developer, none-the-less). Those who were not rooted had to wait a month before it was removed, which was a huge security vulnerability.
Please tell us the carrier you're on, we do like having these conversations in the ATR forums.
Also, more stuff to read:
http://androidforums.com/faqs/443075-root-terminology.html
http://androidforums.com/faqs/443432-what-root-being-rooted-mean.html