Most card details are stored by payment processors, not your own server or a third party shop provider's servers. Paypal and Stripe and various others are probably more trustworthy than anything that stores the details with the shop itself.

But that doesn't really apply to a lot of things. Comments for example, do you really trust a third party more with those? Because if your site is in a grey area, then it's very possible their terms/country/whatever might require them to ban discussion of the topic. Self hosted means your rules, not a large corporation's.

Besides, any middleman is the weakest part of the chain if someone wanted to shut down a site or significantly cripple it without going through a court case. You may like controversy, but a large company would rather see the back of anyone that might potentially hurt its public image. We already see issues where internet mobs go after hosting companies and providers based on something someone said on Twitter. Every third party service is yet another potential target for them, and one that could buckle even more easily than the hosting company (especially if you're not paying for their services).