The pickup picker lives in the cart. A customer who presses a fast checkout button on the product page never sees the cart, so they reach checkout with no point chosen. The "Buy now" button card on the Customer experience tab of Settings decides what happens to those buttons.
The card is one checkbox, Hide the "Buy now" button, ticked from the moment you install, so a fresh install already hides them. It applies to product pages, which is where the buttons cause the problem. Your cart and checkout are untouched. The buttons stay visible when the app is off, when the product needs no shipping, or when the customer is outside the countries the app serves (today Poland and Italy), because there would be no point to pick anyway.
Untick the box and save if you would rather keep fast checkout. The app warns you: "Customers can still choose a pickup point after checkout, but they will skip the map before payment." Hiding the buttons costs you the conversion lift fast checkout gives you, and showing them costs you point selections. Neither is wrong, and you can change it back.
Whichever way you set this, follow Add pickup selection to the Thank you and Order status pages.
If you want a payment method gone rather than a button, use the Payments card. It works at checkout and does not touch product pages.
Check a product page on a phone after saving, since fast checkout buttons are more prominent there.
Stuck on any of this? Open the chat in the bottom right corner of the app and we will help you set it up.