What's the point of asymptotics? I presume this is a rather stupid question, but I hope some of you can find a bit of time to entertain it.
Looking at asymptotic behavior of estimators/test statistics etc means looking at their behavior as sample size approaches infinity. But if our sample size is infinite, doesn't that mean that we already have the whole population?
Case in point: consistency of OLS estimates. Under certain assumptions we are guaranteed convergence of estimates to true parameters in the limit, but at infinite sample size we already have the whole population, so why would we even want to estimate anything when every possible data point is already contained in the sample, all we have to do is to perform a lookup.

The first reason we look at the asymptotics of estimators is that we want to check that our estimator is sensible.  One aspect of this investigation is that we expect a sensible estimator will generally get better as we get more data, and it eventually becomes "perfect" as the amount of data gets to the full population.  You are correct that when $n \rightarrow \infty$ we have the whole (super)population, so presumably we should then be able to know any identifiable parameters of interest.  If that is the case, it suggests that non-consistent estimators should be ruled out of consideration as failing a basic sensibleness criterion.  As you say, if we have the whole population then we should be able to determine parameters of interest perfectly, so if an estimator doesn't do this, it suggests that it is fundamentally flawed.  There are a number of other asymptotic properties that are similarly of interest, but less important than consistency.
Another reason we look at the asyptotics of estimators is that if we have a large sample size, we can often use the asymptotic properties as approximations to the finite-sample behaviour of the estimator.  For example, if an estimator is known to be asymptotically normally distributed (which is true in a wide class of cases) then we will often perform statistical analysis that uses the normal distribution as an approximation to the true distribution of the estimator, so long as the sample size is large.  Many statistical hypothesis tests (e.g., chi-squared tests) are built on this basis, and so are a lot of confidence intervals.
