The vast majority of public shareholders don't vote their shares. A VC is much more likely to apply unwanted pressure to the board/management than the general public is.
IMO, the best reason to avoid an IPO is to stay out of the media.
Harder for activist investors to get into a private company than a public one imho. Keeps out those who would squeeze the business and bail, and potentially kick out the founders. With sufficient cashflow (which Stripe most certainly has), you can buy out existing investors without going public.
(not ex-Stripe, but own startup equity and have no problem with them never going public if that is the choice; optimize for the enterprise and existing stakeholders, not the public market mechanics broadly speaking)
The VC likely already has ownership, and a board seat - public companies are susceptible to activist-investors and hostile bids: outsiders who hold little/no stake, but an outsized influence.