If I were to leave an alcoholic spirit out to evaporate naturally (e.g. Campari) until it was just a powder, would that powder be non-alcoholic when rehydrated? In other words would 100% of the alcohol be removed?
Ethyl alcohol is a liquid at room temperature, and in fact doesn't turn into a solid unless cooled below about -114$^o$C! So, whatever solid you might have after drying your spirits to a powder would would be thoroughly alcohol-free.
Vapour-liquid equilibrium (VLE) of water-ethanol mixtures is very well known. We can predict that the vapour in equilibrium with the liquid contains more ethanol than water for solutions with less than 95.6% ethanol (weight basis). In particular, ethanol is much more volatile than water at concentrations around 1% ethanol (in weight basis). To know the evolution of the water-ethanol solution, you must solve Rayleigh's equation (e.g. here).
If your alcoholic spirit contained only water and ethanol, as it evaporates it would become a solution with an increasingly lower amount of ethanol but, strictly, it would be not possible to obtain an ethanol free solution. However, for any practical purpose, you would obtain at some point concentrations which can be considered in effect ethanol free.
Now, your solution (e.g. Campari) contains carbohydrates, oils... It is very difficult to predict the evolution of the concentration of the remaining liquid as it becomes a very concentrated solution of these components. For instance, we can expect traceshat part of the ethanol will be retained by oil droplets
PS: Mathematically, Rayleigh's equation gives place to a "first-order system" similar to, e.g., a cup of tea that cools down at room temperature. The cup will never reach room temperature in finite time, but after a while, the cup temperature becomes room temperature for any practical purpose. In reality, after a long time has passed, the temperature difference (or the concentration difference) is so low that the hypotheses of continuum that were used to deduce the equations do not hold any longer.