This is a far subtler and deeper rabbit hole than appears. Harrisson, Kreps and Pliska’s papers explain the completeness, uniqueness and risk-neutral aspects.
With regards to the BS framework, it simply states a real-world dynamics in the form of a GBM. The no-arb condition is essential in that it dictates that the riskless portfolio of option and hedge must grow at the risk-free rate, and from there allows the derivation of the BS formula. Note that no assumption or statement needs to be made about the risk-neutral dynamics of the stock price.
Subsequent findings about the necessary dynamics under no-arb conditions and completeness of the market have then yielded the risk-neutral GBM and the fact that the option value is the discounted expectation of the payoff under the risk-neutral measure. Under that framework, one naturally recovers the BS formula from a different angle.
But that latter point is historically unrelated to the derivation of the original BS formula.
In other words, one approach uses the real-world dynamics and a no-arb argument to arrive at the BS formula.
The other uses the no-arb argument and market completeness to derive a necessary fundamental pricing approach (expected value) and the risk-neutral asset dynamics. From these the BS formula can also be obtained.
In my opinion the formula can almost be thought as a law of nature (under certain assumptions on nature). It exists conceptually independently of its discoverers. What BS have done is discover one way to derive it, and subsequent research uncovered another way.