Seriously? Not trying to say that you're lying, but is there a documented example of this happening in the past? I could see why a player would do something like this.
It is rarely made public because it is rarely relevant.
Carey Price is one of the more famous cases where it was publicized because Montreal was in a similar position to Nashville shortly after extending Price. If you scroll down at Capfriendly and look at the previous contract that Price was on, you will see that in the notes section they mention that his modified NTC was turned into a NMC after he negotiated an extension.
I have no way of knowing how common this is, but I would guess that it must be common place. As long as the player isn't finishing up an RFA year where they cannot have trade protection, it only makes sense to have the trade protection upgraded to reflect the trade protection in the extension. Otherwise, a player puts themselves at risk of being flipped by their team shortly after extending if things go badly.