First of all you are confusing statistics with mathematics. In mathematics we operate with certainties and there you are quite correct that all that we can know with a sample size of one is that all we actual know is that there is at least one thing to sample and at least one of the entire population has the traits of that sample.
But you see, that is where you have effectively argued that all sampling is meaningless. If there are a thousand black sheep on the field in Scotland all you know is that there is at least one field in Scotland, it has one thousand sheep and they are black on at least one of their sides. And so on. When you operate with certainties you will never know before you have sampled the entire population.
In statistics however you will try to quantify the different uncertainties. In science we can then use this to set certain thresholds to circumvent the problem that you often can only disprove certain things and never prove them. Basically all that we can know is that we are*, which makes it very hard to build a scientific foundation on.
Back to our sampling. I have never claimed and will never claim that if you take one sample that whatever qualities that has is the only one present in the population. To claim that I did by arguing against it is quite frankly something you should consider yourself too good for. What I am saying is that it is highly unlikely that you have found a unique entity in the population**. And this is what I mean with resolution. The bigger the sampling size the better resolution you get on the picture it casts on the entire population.
*Cognito ergo sum
**With regards to certain attributes. All humans are unique when you come down to it, and thus by selecting any individual out of the entire population you will never find any one equal. But that is not what you are after when you look at groups. You have to look at traits that are similar. Like how many legs, skin colour, reads the Register and so on.