I am trying to explain/understand how one gets from sample observations to making inferences about a population parameter.  I want to make sure I explain the logic correctly.  Let's take the most beginner example where we have a normal population, a known variance ex ante, and we want to make inferences about the population mean.

When learning, a lot of "properties" are thrown around like CLT, law of large numbers, etc... I am hoping to focus on the most relevant properties.  After re-reading a few times, seems like CLT, the fact that the sampling distribution of the sample mean is normally distributed, etc... are just intermediary properties (basically steps within a proof) - not the property that allows us to ultimately make inferences.

What really matters for hypothesis testing is the test statistic itself.  The test statistic *relates* our sample estimate and some value.  Usually, we like to substitute what we think the population parameter may be for the value.  In this case the test statistic is the Z-statistic.  For example, when we look at this:

$Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$

When it comes to interpretation, what we should be thinking is that given $\bar{x}, n, and \sigma$, the expression above gives us the likelihood of $\mu$ being certain values.  The likelihood is given by the standard normal PDF.

So I guess this brings me to my question.  If I wanted to do significance testing for any population parameter, would this be adequate steps?

 1. Determine a way to estimate the population parameter using sample data.  Usually, do we need our sample estimate to be unbiased?  What happens if we can't find an unbiased estimate?
 2. After doing that, find some sort of expression that relates your estimate with the population parameter so that that expression follows a well known distribution.
 3. Assuming the test statistic follows a continuous distribution, the test statistic will give us the likelihood of our population parameter taking on certain values.

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I know this is a bit overthinking, but when I learned this stuff, we just learned how to make inferences with the Z-statistic via calculate some formulas and then look it up on a Z-table.  I just tried to explain it away by saying some stuff like the normal distribution is symmetric and we are basically standardizing some stuff, etc....  I never learned to view the test statistic as an expression that tells us, given sample data, this is the likelihood of our population parameter being specific values.  As a result, there was always a gap in logic as to how we actually are able to make inferences about the population using our sample.