It's hard to parse this question. Who is the "I" in this question? And when is the time in question? An almost trivial answer is finding a location/scale family that is $\propto \exp(-x^2)$. The OP then goes on to ask "If humanity forgot about the normal distribution, in what manner would it be rediscovered"? This is an altogether different question. I think a relevant answer here is one that 1) borrows the perspective of modern science 2) provides an answer that is different from the most frequently encountered historical answer, aka the Central Limit Theorym.
In quantum mechanics, information theory, and thermodynamics, the entropy quantifies the state of a system. In these fields, the quantum state is in fact, wholly random or stochastic. Contrast this with classical mechanics. In classical mechanics, states are fixed but our observation is imperfect due to the contribution of hundreds or millions of unobserved influencing factors: this kind of result gives rise to the CLT.
In quantum mechanics, we use Bayesian probability to quantify our belief about the state of the system. Along those lines, proofs have been presented, and tweaked, that the Gaussian or normal random variable has maximum entropy among all random variables with finite mean or standard deviation.