This one divides people ;-)
There is a very simple answer to your question; usually proposed by people with decades of markets experience, for whom (a) a Monte Carlo proof of statistical consistency is "enough". And (b) who tend to think that market uncertainty will always trump model uncertainty many times over. Which can upset a different group, who get upset by the lack of greek letters, associated lengthy calculus. and formal proofs. The difference is more philosophical than substantive; because the two approaches don't tend to suggest very different outcomes when applied to real-world data.
It comes down to whether you are happy making the following intuitive statement, or not. "The significance of Sharpe>0 is the same as that of Returns>0 given Time and Volatility". Assuming a large sample (and thus Student's T ~ Normal Z, as you you say), then:
Returns = Sharpe * Time * Vol
Timed Vol = Vol * root(Time)
The one-tailed p-value is Inv-Normal(Returns/Timed Vol), equals N'(SR * root(Time)). Simples...
Many (often more scholarly) commentators are not happy with the initial intuitive assumption above. IE P(SR>0) = P(Returns>0 | Vol). They do not think of the Sharpe as a convenient ratio for comparing different securities; but as a phenomenon in its own right; with its own distribution. In which case, they would argue that it has its own distribution and its own standard error, in its own right.
As opposed to volatility already being the standard error for returns; and Sharpe being returns/SE, equals already the Z-score (or T-stat) for the simplest hypothesis test as per Stats 101.
Whichever logic is "correct" depends on my intellectual and practical priors here, I suppose ;-)