Is normality testing 'essentially useless'?

A former colleague once argued to me as follows: We usually apply normality tests to the results of processes that, under the null, generate random variables that are only asymptotically or nearly normal (with the 'asymptotically' part dependent on some quantity which we cannot make large); In the era of cheap memory, big data, and fast processors, normality tests should always reject the null of normal distribution for large (though not insanely large) samples. And so, perversely, normality tests should only be used for small samples, when they presumably have lower power and less control over type I rate. Is this a valid argument? Is this a well-known argument? Are there well known tests for a 'fuzzier' null hypothesis than normality?