First of all, it's worth noting that Summoner typically hasn't been a busy caster. It's busier than BLM, but as of SB despite the tedious micromanagement of contagion, it was never a busy caster. Red mage was always the "busy" caster and it has a playstyle that best suits a busy caster. That is, half of all GCDs, always, are instant cast. So it has a lot of opportunity to make use of oGCDs, which it has many of, though ironically nowhere close to the new Summoner.
After this, the classes "busier" than Red Mage are always the Ranged DPS classes. Bard, Machinist, and now Dancer. Of these, Dancers appears to be the least busy, but it's still moderately busy, and I believe Bard is the most busy as of ShB. So if you want a busy playstyle, the playstyle exists both in caster form and in other DPS, including melee.
What breaks it for summoner is that the class effectively doesn't have enough opportunity to do what it needs to do. At level 72, you basically have enough of the rotation to know everything if you never enkindle or use EAs during Phoenix, it's basically the same just mix in an Enkindle Phoenix oGCD and you'd get it.
The classes breaks in about 3 areas. 1, the opener has literally about 18 oGCDs it needs to cast within 15 seconds. It needs to double weave but if you run into ping issues, double weaving is likely beyond your capabilities to reliably do. I've noticed some SMN oGCDs double weave worse than others, making the entire issue more frustrating than it needs to be as well.
The second is that the nature of how EA works means it is very easy to get the game to throw errors at you and eat the cooldown without giving you anything to show for it. Or maybe it does and the error is erroneous, but it throws out errors like mad during normal play.
The third is that the AoE rotation, due to how busy the ST rotation is, is so problematic that you flat out cannot cast Outburst outside of DWT because you don't have enough time and are flatly inundated with too many oGCDs to cast, so the rotation especially in AoE is constantly spent either not casting anything, or casting ruin 2/4 to get out your ESes, your Painflares, your banes, and your EAs. This isn't even factoring in Enkindle and Devotion, which you still have to find room for.
Play the class at level 60 and, despite it having some of the worst offenders of the rotation around, you'll notice the class flows a lot better overall, albeit the aoe rotation is still a complete and utter nightmare.