Does a fully-resolved phylogenetic tree have to be dichotomous?
In theory, yes, every tree has to be dichotomous. You can understand a trichotomy in a tree as the summatory of two dichotomies that had happened so close in time that you cannot know wich was first.

Given a certain population, assume that some individuals colonize a new environment, got reproductively isolated and form a new specie. This is the typical speciation process. In this case, the two species, share a common ancestor. If you go back in time, all the individuals of the new specie will descend of only one individual. The same goes for the original specie. If you go back further, this individuals will share an ancestor, too. This is the conceptual meaning of the dichotomy, but normally it is impossible to determinate wich was exactly the common ancestor.

Now imagine that the original population formed not one, but two new species. And that this process happened about the same time as the one described before. It is conceptually possible that a set of three brothers were the origin of three different species (wich will be a true trichotomy), but not only it's very unlikely, but it's virtually impossible to prove. Since two close dichotomies are far more probable than a trichotomy, it's assumed that every tree has to be dichotomous, and that a trichotomy is in fact due to lack of resolution.