They've expanded into new countries, into mobile with both apps and Square-style card readers, into POS with at-register integrations in store chains, into identity management, into lending... if you think they're stagnant, maybe you haven't been paying attention to them.
Whenever I hear terms like "vertical-oriented", I'm reminded of the managers at the big stagnant corporation I used to work at. Those terms sound just like "value proposition", "leveraging assets", and so on. They're not talking to developers or users, which are the people they need to focus on.
Case in point: I wanted to use PayPal to let users buy my apps, went through the x.com site, and quickly dropped it. Way too complex. Now Stripe on the other hand, that's something I'd really like to integrate with my apps.
I'm not saying PayPal can't possibly improve the situation, but they need to start talking to their customers in plain, simple language. What they're doing now is a red flag to me that I've seen in several other places, right before they were obsoleted and fizzled.
Just because they're not iterating in the dev-oriented U.S.-centric C2B space, the least profitable and easiest to solve payment spaces, doesn't mean they aren't innovating. To put things in perspective JANA, arguably the largest payment platform in the world by users, is virtually unknown in Silicon Valley.
But to take one of your examples, they started offering Square style card readers after Square did. So at that point they're just catching up with the market.
Have you tried purchasing a typewriter lately? The new market is almost nonexistent, and the used market is a mess! This is definitely a market due for disruption!
A well-made typewriter is currently worth more than most startups. Indeed, even a poorly made typewriter is probably more useful than all of the social networking, kickstarter, and faddish clones currently flooding SV; it will certainly provide more value to its user and society.
https://www.x.com/products
They're also doing new vertical-oriented products for governments, banks and education.
https://www.paypal.com/webapps/mpp/other-solutions-overview