“Any measurement in an inertial frame of reference is as valid as any other.”
Of course, when you are doing the math. Yet, when it comes to reality, it’s still quite obvious that not all reference frames will represent an accurate picture of objective reality. For example, if you were moving at the speed of light towards Alpha Centuari, the distance between the sun and AC shrinks to ZERO, from your perspective, but still, the two stars would not collide with catastrophic consequences. So while your reference frame would be completely valid mathematically, it is NOT as valid a representation of reality as just about any other reference frame you could pick.
Your example doesn't apply, as special relativity does not allow light-speed travel. (The FTL travel of the article involves stretching distances behind the traveler, so zero separation again does not apply.)
One could point out that near-light-speed travel would bring the stars into arbitrarily close proximity. But many other observations would also change, e.g, the shape of the stars; if one works out ALL the physics, no laws are violated in the traveler's frame of reference.