I don't think this is true. If you have Verizon for your carrier, for example, it doesn't matter where you bought your device, from Verizon or Google unlocked, the updates aren't available any sooner for the unlocked version because ultimately the carrier has to approve before you will get it. My fiance has a Google purchased P7Pro and I got mine from Verizon, yet neither of us got the update via OTA until yesterday afternoon, when everyone on Verizon could get it. The version you have from Verizon on my Verizon purchased device is exactly the same as someone who bought the same device from Google. I've had Pixels since the OG 1XL, all purchased from Verizon, and I know others who have the same devices from Google and I always got my update the same version and day as them. The updates come from Google initially, but it is the carriers that "allow" them to go through to their customers regardless of where they bought their devices.
If you have an unlocked/unbranded phone, it can easily update completely outside of the carrier network, if you simply remove the SIM, and scrub any other carrier metadata, it'll update like a brand-new phone, out of the box. At this point, the carrier cannot gate it, as it's not part of their network, you can easily just update and move to another carrier, without issue.
If it's carrier-branded, your model is slightly different (mostly the locked part of the ROM), and their build deployment will be slightly different as well, in most cases. The carrier completely gates it in this case, and you cannot move the phone to another carrier, or (carrier) unlock it, until they issue you a code to do so (typically after payoff and a period of time, you can request an unlock).
Yes PGrey, I agree with you on that scenario with SIM removal. I didn't think about that when I wrote my response as my intent was to correct the statements that if you buy from Google unlocked, and put it on Verizon (or other carrier) that somehow you get the updates differently than those who bought the device from Verizon.