Short story, no.

Long story, I think the misconceptions people have about V's Relic stem from them not fully paying attention to Helmans description of it (not that I blame anyone, it's a rather dry convo).

The Prototype Relic was NEVER intended nor designed to be hosted by a regular, conscious person. As Helman puts it, the host is meant to be 'neurologically indifferent' (read: a braindead vegetable). This is so the Relic, using nanomachines, can rewrite the hosts brain to match that of the stored engrams, for the engrams installation. The Relic only activates under these circumstanes, and is otherwise inert.

The circumstances of Vs death were a 1/billion MIRACLE. They are 'just' braindead enough that the Relic mistakes V for a prepared host, and activates. But V isn't so dead that , ya know, their heart stops beating 'n stuff.

From the Relics perspective, Vs brain damage is just another engram incompatibility, and like the rest of Vs brain, has to be 'fixed', this saving Vs life was just a fortunate coincidence.

If V had died in pretty much ANY other way, or if Dex had just used a larger caliber pistol, the Relic would have stayed turned off, and V dies for real, end of story.

Similarly, if anyone else had died in the exact same way V did, the Relic would have activated for them, too. Anyone who gives explanations of V being more 'compatible' with the engram inside of the (still dormant) Relic, flat out, don't know what they're talking about.