You can see any requests your browser makes, and this is a load of nonsense
It's possible for anyone to see HTTP requests in their browser by using the Developer tools (F12 in Chrome) and then opening the Network tab. Any data which is being sent is shown in the Request section. This is the case for any website and any request including ajax requests which are commonly made through JS libraries.
So, all these people moaning have tried it on a site where Stripe.js is present on every page? What did you see in the Request data that "logs all your activity"? Oh, you haven't, and it doesn't. I know this because I've done it myself.
It does send some data aside from for the payment, including a timestamp of how long you were on the site. But that is legitimately useful in combatting fraud as it could identify a bot submitting card data, amongst other things such as geolocation details.
I've already replied on a previous comment where someone was trying to describe how they thought JS was to blame for the BA attack. A total lack of understanding of how that actually worked (I've explained it in my reply).