Firstly, the degree of marketing bullshit that you have to push through just to get to the truth of the APIs is staggering.
Secondly, they perform backflips to avoid telling the truth about their integration options, which is to say that they're all fundamentally insecure for real time digital goods and services unless you implement IPN plus polling plus duplicate detection / additional round-trip validation calls, ie. you can't trust an IPN notification. The degree of complexity (order state tracking, IPN state tracking and duplicate detection, retry support, running your own IPN-receiving server) and latency requested here of client businesses is immense.
Thirdly, their idea of international support is pathetic. It seems that they've basically duplicated their entire business process to other countries, translated it, and then assumed that everyone in that country requires only one human language in all of their interactions: documentation, support, emails, etc.
Finally, as is widely known throughout the industry they have a shocking reputation for the arbitrary suspension and seizure of accounts and assets, with little to no recourse for those affected.
I am not surprised that Stripe has taken off. Unfortunately, that's still lipstick on a pig: fundamentally, the settlement, risk, transparency and government interaction model of credit card networks means they are unsuitable for an increasingly large volume of business around the world.