Alternate statistical test to linear regression?

First, please expunge the "(N>30)" concept from your mind.  This is an
oversimplified rule of thumb used in introductory statistics courses
(I am guilty of doing this in intro stat as well, but I try to
emphasize to my students that it is only a rule of thumb for that
class and the truth is more complex once you are in the real world, so
consult with a statistician).  There is nothing magical about a sample
size of 30, I have seen cases where n=6 is large enough for the CLT
and cases where n=10,000 was not big enough.

If the data is not overly skewed and your sample size is large then
you can just use regression as is and the inference will be
approximately correct (with a really good approximation).  But with
skewness we often prefer the median over the mean and least squares
regression is equivalent to fitting a mean, some of the robust
regression options are equivalent to fitting a median, so they may be
preferable on that count.

Note that Pearson's correlation does not test linearity, it assumes
linearity (and bivariate normality).  Most issues with regression will
be the same for the correlation.

Alternate statistical test to linear regression?

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