It's generally transparent redirects in the other direction. You load JS from their servers, the form posts to their servers, redirects back to yours. Often even the form isn't hosted on your infrastructure, but instead embedded into your page through some JS that is hosted elsewhere.
If your responsibilities weren't waived or significantly lessened, I'd imagine companies like Stripe would be significantly less successful.
You generally don't need to load any JS from the payment processor to use a transparent redirect, although I'm sure some work this way. Transparent redirects just require setting the form action property on your payment page to a URL on the processor's site. The processor then silently redirects back to the next step in your process.
Using an embedded payment form also shouldn't require any JS, as it is usually done using an iframe. This method should be safe, as the same-origin policy in the iframe would prevent JS on your domain from interacting with form elements in the iframe. But this is not typically what people are talking about when they talk about a transparent redirect.
If your responsibilities weren't waived or significantly lessened, I'd imagine companies like Stripe would be significantly less successful.