As far as interonnectivity is concerned, WHM has Asylum and Temperance that provide healing buffs and Plenary Indulgence which adds potency to Afflatus Solace and Afflatus Rapture.
AST can use Helios and Aspected Helios to give Horoscope more potency and a longer duration, Neutral Sect to get a healing buff and to add a shield component to Aspected Benefic and Aspected Helios. Lastly, Synastry can be used to share a portion of the potency of your single target healing spells to another target.
SCH can use Protraction, Fey Illumination and Dissipation to get healing boosts, Recitation to force a crit on some oGCDs and GCDs healing skills and save resources (be it MP or Aetherflow). Lastly, it can use Emergency Tactics to convert the shields from Adloquium and Succor to raw healing and Deployment Tactics to spread the Galvanize effect of Adloquium. It was rather funny to see that you described Sage as complex and SCH as "intro healer" because a popular complaint about SCH is that it allegedly needs to use more buttons and all these "gigabrain combos" to do what AST and WHM do with one button.
SGE's interconnectivity is not exclusive to it and it has a rather straightforward and intuitive kit with little to no caveats attached to it. The biggest "annoyance" would be the fact that Taurochole and Kerachole can;t stage their mitigation. AST has some tricky tools to use (Earthly Star, Horoscope, Macrocosmos) that require more awareness to use properly while SCH has to deal with some caveats like Dissipation being the only healer skill that can actively punish you if misused, several abilities not working together and so on.
As a whole, SGE really doesn't even enough on its plate to be considered 'complex' (which is not inherently a bad thing), not only because of FFXIV's low bar, but also because there are at least two healers this term would fit better.