This is unfortunately the common view. And many of the tanks I see here are rather coral collections than naturally looking biotopes.
To me recreating a natural looking biotope is the highest level of a reef tank.
I'm in the process of growing a reef to provide my fish a natural and intellectually stimulating environment. The fish are the stars and the corals the decoration. Of course you need to match both to be compatible.
Think of it like living in a nicely decorated house versus living in a barren shed.
Fish that live in a tank that largely resembles their natural environment will behave more natural, have brighter colors and be overal healthier.
Also most underestimate the intellectual and social stimuli fish need to not just survive but thrive. For that reason I aim at keeping all my fish in pairs or social groups.
The fish you have selected can to my knowledge be sexed and kept in pairs which will give you an even more special tank and interesting social interactions (single lion fish can get boring very fast).
Fuzzy Dwarf Lionfish Dendrochirus brachypterus
How to Identify a Male Lionfish From a Female
Some eels alter colors; other morays, their sex
If coral means "stony coral" than this is largely true. But coral has a much larger meaning and there are quite a few that will not only survive but actually thrive in "dirty" water. Actinia anemones work great in a fish tank and some clownfish like A. clarkii even accept them as hosts.
There are also quite a few zoas (usually the ones not in high regard among the fancy zoa collectors) that do well with less than perfect water quality.
And then there are all the other inverts like feather duster worms and crabs.
Certain cleaner shrimp like to hang out near moray eels which would make a nice biotope.
I would not forgo inverts with fish like the ones you are interested in. They are actually more reef safe than most fish people keep in their reef tanks.
As for biological filter and skimmer:
Think of the bio filter as the liver of your tank that transforms toxic metabolites (Ammonia, Nitrite) into less toxic (Ammonium, Nitrate) while the skimmer is the equivalent to your kidneys that actually remove food leftovers and metabolites before they decompose to toxic substances. But the skimmer is also what you could consider the lungs of your tank and improves the gas exchange tremendously.
A bio filter is not required if the tank is set up with a lot of live rock - that will be enough of a bio filter in itself. In this case you may just want to have a mechanical filter to remove particulates, leftover food, poo,... Of course that filter should be cleaned frequently, ideally daily.
A skimmer is something I would never want to skip on, even if it is only for enhancing the oxygenation of the water.
