There are cases where you would not want to have a skimmer or only minimally skim, such as if you were trying to keep lots of filter feeders like sponges, sea squirts, etc, doing a seagrass biotope... However, such a tank is not recommended for beginners, you have to keep a very, very low fish load, and you can't keep anything that needs very clean water - almost all hard corals, many softies, clams.... The skimmer is not a backup to a refugium but an additional component, removing DOM before it begins to be broken down by the biofilter in your refugium and main tank. And filter-wise, a refugium has nothing unique that is not also occuring in your main tank - it adds more LR
to your tank, serves as a place to grow macroalgae to remove nutrients when you prune back but you could also grow macro in your maintank depending on what fish you keep. Many people consider a skimmer at the most important component of a reef tank or fish-only, and I think the only case where you might not want one, apart from the more advanced filter-feeder tank discussed above, is in a nano-reef.