It would complicate things too much if there is no thulu. Also, Cthulhu currently acts as a solo cult member (it can actually solo win by being the sole remaining player).
The reason people don't lynch thulu in sandbox is that thulu usually doesn't even have its own cult, although if cult were known as well as thulu it would definitely be lynched.
Thulu+cult might be somewhat powerful, although cult is countered by masons, cl not having a claim to fall back on (easier to find than other members), all intercepting roles, and the fact both mafia and town want cult dead and can kill them at a faster rate than they can cult.
Another simple edit would be making it so that killing Cthulhu cures all thulhu'd players but I think it's fine how it is.
I don't think it's a terrible idea, it's just that for it to be implemented you'd have to get lucid to code it. Also I think that if someone visits the thulu, they're easily outted as CL and and if thulu is lynched then it's autoloss for the cult team on d1 and i think that comparatively puts the cults at a disadvantage than their current position in games if you look at it from a ranked point of view. It's still a pretty neat idea though and I like that there's some discussion/debate about roles strategies etc on the forums as well
There would be motivation to out Cthulu, unless the person who found cthulu felt cult was in a position to win, and wanted to be culted. Then they could out a fake Cthulu, or else not out anything at all. Of course it would make finding cthulu in the early game easier, but cult is already pretty overpowered for most setups anyways.
I feel like there was already a role or icon or something made for the actual cult leader but I don't know much about it.

I'm assuming Cthulhu would have to participate in cult meetings then and in setups where there's already one Cult and one Cthulhu like Lovecraft Rising, the original Cult member would just act as a normal culted player.
One problem is that killing the cult would be extremely easy as soon as one person visits Cthulhu. Players can currently point them out by spamming single capital letters into the chat, spelling out Cthulhu’s name.
And in setups where there is no Cthulhu like the party setup, the mechanics would have to fall back on what they currently are or the setups would have to be remade, subbing in Cthulhu for the original Cult.