As a Swain player it's quite simple really.
Swain, even the old version, had a very specific play pattern that is not typical for mages. That is to be close to the target, tagging them with spells, and then all in with your ult (not that different from an assassin really, except your mid game goal is to become a bruiser not a glass cannon).
So, he is what they define as a "battle mage" through and through, but has a unique play style. There isn't much reason to pick him for burst/crowd control because there is always someone who does that better. He sustains quite well, but so does Vlad if that's all you wanted was a safer laning phase.
His dedicated mains mostly dropped him after the rework, then he was temporarily popular when he was strong; now that he is nerfed to being balanced he doesn't have a solid enough niche to choose him over something else. I feel the loss of a lot of old Swains fun mechanics, but I connect with the thematic and play style enough that I keep playing him anyways.
The old version was less healthy for the game in that he had point and click DOT's, but had a defined and powerful niche. The new one doesn't have as much crowd control or defined strengths.
In summary think about it like this...
Want more reliable burst on an immobile mage?... Syndra, Velkoz
Want better sustain?... Vlad
Want higher amounts of crowd control?... again Velkoz, Syndra, could debate Veigar with his stun cage, Lissandra, Malzahar
That's just considering immobile mages. Then you got things like Galio who is mobile, comes with a shield, and still has the crowd control and burst.
Swain is completely still playable and fine... but "fine" isn't strong, or makes you want to pick him up for anything. I still like the character and will continue on with him (been playing him for over 3 seasons now), but I totally get why he doesn't show up that often.