Yaarel
🇮🇱He-Mage
Why the use of reaction? Any other save doesn't use your reaction and what happens if you've already used your reaction that round?
The reason for the immediate reaction.
The first death save determines whether a disabling injury occurred when reaching 0 hit points − the sword stab thru the gut, sotospeak.
It doesnt really make sense to see if the death blow happened several rounds after the sword thrust.
Moreover, the immediate reaction to make a death save will make players more anxious about reaching 0 hit points − as they should be − because it might inflict a wound that cannot simply be shaken away next round.
That said, your point about the reaction being possibly used already is fair enough. The intent is that the first death save happens at the same time as the lethal strike, and determines how disabling that strike was.