I have successfully published a (very simple) Instant App linked to my personal website.
When I click on my website's link, I was expecting the app to start directly, but instead, I first see a screen with 2 buttons: "Open app" and "Open in browser":
Why?
Other instant apps that I tried do not show this screen and open the app directly (the desired behavior).
We call this screen the "speed bump". It's a security feature. You're right in noticing that it doesn't always trigger. But there's nothing you, as a developer, can do to control whether it shows. The intention is for it to be a black box from your perspective.
We're open to feedback if that's problematic for your app.
This is designed in my opinion to happen at least once to give the user an opportunity to decide what to use: either browser or the instant apps feature.