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)
You'd need to amass 50% of the shares to kick out the founders. That'd be impossible for a hostile party to do if Stripe IPO's because they wouldn't release anywhere close to that number of shares.
The only way to kick out the Collison's would be for the VC's to do it. They currently own 80%. It's easier for the VC's to do that if Stripe stays private than if Stripe IPO's.