Suppose we have $K$ subjects and a treatment with two levels, "Before" and "After". A paired t-test is equivalent to fitting a fixed effects ANOVA:
$Y = Subject + Treatment +\epsilon$
It is also equivalent Repeated Measures ANOVA or Mixed ANOVA, where Treatment is fixed and Subject is random. It tried all four methods for this simple dataset and the Treatment p-value is exactly the same. For the sake of this question, let us consider the fixed effects version only.
The total number of observations is $2K$, while the number of parameters is:
$ p = intercept + (K-1)$ subject effects + (2 - 1) treatment effects = $K + 1$
That is, there are about 2 observations per parameter for any $K$. To me it suggests that the model is likely to be very overfitted, unless there is a very substantial subject effect.
In practice, how often have you seen the subject effect so large that it justifies pairing the observations by subject?