The difference is between the company having their own merchant account with a bank (which is what most large companies do) using an online payment gateway, and not having one and leveraging the processor's instead (which is what Stripe, Paypal, etc provide). When you apply for a merchant account you get that approval and underwriting, but with a hefty application fee for obvious reasons. If your payment gateway shut you down, you can just switch to a different one, but there'd be little reason for them to do so. Your bank is much less likely to shut you down, because you were preapproved. The main reason would be for high fraud/chargeback percentages.
When you use Stripe or Paypal or similar, you don't apply for your own merchant account. You make transactions using their merchant account. If there's a fraud or chargeback percentage issue, the banks will have a problem with them, not you, but it also means the service needs to be proactive in policing their clients so the banks never come after their merchant accounts.
When starting up a company, use a Stripe or a Paypal to get up quickly, but probably ramp up to using multiple quickly, so you have backups. As your revenue increases, apply for a merchant account and move your transactions over to that. There is an upfront cost, but the processing fees are significantly cheaper, and no one will pull the rug out from under you without quite a bit of correspondence. Even when using your own merchant account, you can find processors who will handle all the credit card input and transmission on their end instead of on your site, which greatly limits your PCI compliance requirements. Regardless, when you build your service, abstract the payment process such that you can easily add or switch providers. Don't be married to a single one, because at the least you should be switching to a merchant account when the application fee is lower than the transaction fee percentage difference.
Source: I also worked for (and was the principle developer of) a high risk payment processor, providing a processing gateway for individual merchant accounts serviced by an ISO. We tried to look at becoming an IPSP (I think that's the acronym), letting customers leverage our merchant accounts like Stripe or Paypal do, but it was significantly more work and process with credit card companies than we wanted to deal with.