Giovanni's being the Viridian gym leader insinuates a possibility of him sharing monumental backstory with Lance. After all, the champion (given they're not a minor) is heavily implied to oversee the League, and as such, must know what happens to the money in the organisation. They can't just know where the funding goes, but must also be aware of who they're paying; and as such, they very likely have the power to fire or relocate leaders. At the very, very least, Lance knew the eighth leader's identity.
So what? Well, regardless of whatever protection racket Giovanni could have potentially organised to keep the job, Lance is willing to hyper beam those who get in his way. If he couldn't tolerate Giovanni's holding the position, he'd have long evicted him. Additionally, Giovanni would have been very unlikely to need the income that gym work would garner. Nor would he have had much time to instruct and battle younger trainers. As such, Lance may well be responsible for keeping Viridian under Giovanni's watch. This is a strange consideration, given Lance's quest to extinguish Team Rocket's embers with his own two hands in GSC. If he were complicit in facilitating an association between Team Rocket and the Indigo League, much of his motivation throughout gen 2 is presumably remorse. This remorse isn't just for his part in granting the enemy, so to speak, a safe place and secure salary, but also remorse for not doing more to fix Giovanni, save Giovanni, while he could.
(Part of his motive would also be a dire need to create plausible deniability about keeping Giovanni employed: By defenestrating Archer, Ariana, Petrel, and Proton, he'd have engendered a reputation as a hero, playing into messianic archetypes.)
What I'd love to see in fanfic is some extrapolation of the strange friendship that Lance and Giovanni would have entertained: the dynamic of a mafia godfather and a pop culture hero with an unadulterated mutual respect. Despite their media portrayals as moral absolutes, as the brave knight and the wicked emperor, how much do they have in common? Just as Giovanni earned every grunt and every admin's greatest esteem, what did he do such that Lance would risk everything for some sentimental shred of trust? When did they meet, and how deeply did Lance grieve as he split up Team Rocket? How much love and resentment and withered hope for Giovanni did Lance channel into his treatment of Silver? Did he even recognise him as Team Rocket's wretched heir?
