In short, if you do not use Cloudflare, then access to your website from things that are not browsers is not possible. Specifically, that’s caused by this security system: Ensuring only web browsers can access your website
However, if you use Cloudflare, then it used to be that this system was bypassed (because Cloudflare also provides protection to your website). But that appears to have changed.
As far as I can tell, this was a deliberate system change. Handling these webhooks should not have worked in the first place, and that has been “fixed”.
And as the article linked to before says: the security system is enabled on all accounts and cannot be disabled. So no, there is nothing you can do to change that.