There are a lot of issues with that assumption, but I'll go through it all as concisely as I can.
Oxygen isn't the only limiting factor to arthropod size
Arthropods are exoskeletal by definition. Ultimately, that means that they are an organic creature inside a hard shell, more or less. By comparison to endoskeletal creatures, that means that their outer carapaces have a limit on how hard and strong they can be as they scale up which is less of a limitation on a creature with a skeleton on the inside. If you're dealing with a 'super-earth' that implies higher gravity, meaning arthropods (especially on land) won't be able to grow beyond a certain size as their carapaces would collapse, or their internal organs would put too much pressure on each other lying in a pile inside said carapace; take your pick.
It's volume of oxygen that counts, not percentage.
A planet with a thicker atmosphere doesn't need as high a percentage of oxygen to support earth based life because the amount of oxygen we breathe (and that we need in each breath) is determined by Partial Pressure, which is a fancy way of saying a set amount of oxygen, regardless of the ambient pressure. The Apollo missions flew with pure oxygen environments because they were only pressurised to less than a third of sea level pressure. Conversely, deep sea divers use much thinner mixes of oxygen (filling the gap with an inert gas like Argon) because at pressure, they're breathing in so much larger a volume of air with every breath. So yes, you need a LOT of oxygen (in terms of volume) to support large arthropods, but don't think of that as a percentage as such, and in an environment with a denser atmosphere the pressure on the carapaces might become the limiting factor rather than oxygen (see above).
Your plants must be in place for millions of years prior to animals.
Your planet still needs a Great Oxygen Event in some form to release the molecular oxygen and put it into the atmosphere. Planets are highly unlikely to start off with such an atmosphere because oxygen reacts with so many other elements. That's what makes it so useful as an oxidiser. So, you need a constant endothermic reaction like photosynthesis going for millions of years before animals even come onto the scene, just to supply the oxygen needs they'll have. By the time your planet is ready for animals, it's probably not an archipelago anymore. Just saying.
This is all based on our current model of life
For all we know, on some of these worlds you could have photosynthesizing animals. Arthropods could have lungs, too. They may not have, but could have a form of gills that extract the CO2 out of the atmosphere instead, harnessing sunlight to form their own oxygen. They could have a combination of exoskeletal and endoskeletal body structures that allow them to grow bigger because of a segmented carapace and their growth model could be adding new segments as they grow. That's the thing about evolution; we really don't know what life would look like on another planet because we don't know the starting structures and we don't know the pressures in the environment that force it to evolve in order to survive.
Put simply, the existence of archipelagos and photosynthesizing plants don't in any way increase the chances of large arthropods on your planet, although there is the chance that they make the environment favourable for them to evolve. But, evolution and planetary ecology is an immensely complex and fragile interaction of billions of factors and as such, it would be wrong to say that the one follows the other as a general rule as defined in your question. It's not to say that you're wrong; merely that the process is almost infinitely more complex than that.