I agree with everything, except the bootloader part.
A Verizon S3 will have an unlocked bootloader, so, yes you can root it.
A Verizon S4 (on 4.4.2) will have an unlocked bootloader, so you can root that too.
Same goes for the Verizon S5 and Note 3.
However, on a Verizon Note 4, or S6, or even S6 Edge. Its currently impossible to unlock the bootloader and get root.
As a security measure, AT&T and Verizon have started shipping their new devices with Locked bootloaders that can't be unlocked.
With the launch of the new Note 5 and S6 Edge Plus, Samsung announced that KNOX will be deeply integrated into the kernel and ROM, and the bootloader will remain locked and if any attempt to root is made on the device, it will not boot. Samsung did this as an encouragement for AT&T and Verizon customers to use Samsung phones as 'Work phones', because the updated KNOX thing has very secure encryption techniques.
So currently only T-Mobile and Sprint devices can be rooted, because they're shipped with unlocked bootloaders.
Sent from my SM-N915FY