An earlier MO question highlighted Giuga's Conjecture:

A positive integer $n>1$ is prime if and only if $$\sum_{k=1}^{n-1} k^{n-1} \equiv -1 \pmod{n}$$

For example, for the prime $n=5$, the sum is
$$
1^4 + 2^4 + 3^4 + 4^4 =
1+16+81+256 \equiv 1+1+1+1\,(\rm{mod}\, 5)\equiv -1(\rm{mod}\, 5) \;.
$$
My question is: Is this conjecture in some sense *central* in the pursuit of
prime properties
(as is—surely—e.g., Yitang Zhang's and James Maynard's advances related to prime pairs),
or is it rather perceived as peripheral to the general understanding of primes?
I ask not only because of the spectacular recent advances,
but because the *iff* nature of the conjecture seems almost too good to be true,
even though it has been verified out to $10^{13800}$.