This sounds like a good theory too. I'm not against angels being in this story but i want to be a softie and believe Mai just loves Lily so much she couldn't fall for the devil powers.
The true love exemption for charm and mind control is a classic, and for good reason.
What sort of things a human who sign the contract will get? Money? Power? Anything he wants? In return, what does the succubus get by signing a contract? She can have sex with anyone anyway, or if she likes, she can just stick with one single human for life as if there were a contract.
Traditionally, deals like this are fairly straight forward. Mortal gains service of the devil for a period of time, often until their death. Once that period has elapsed, devil gains ownership of mortal's soul. The services a succubus can render are mostly pretty obvious, but the devil is in the details of what the contract actually stipulates.
It honestly might be as little as 'perpetually attractive wife who's DTF 24/7' with provisions for having children. I could see there being wiggle room for requesting tactical use of devil powers to grease the wheels of career advancement, but I doubt the main focus of the contract is in that direction. Unless succubi are the only devils around, I suspect there are other kinds who make avarice and ambition the primary concern of their own contracts.
As for what they get out of things besides free and clear feeding, various depictions have either souls as a useful personal commodity for devils, or in other cases the devils are operating under a quota with regard to mortals consigned to hell by their actions and may face pressure from infernal hierarchies to deliver them.