The "Build" button and Generative BI features are part of the QuickSight Author Pro license, which provides advanced authoring capabilities.

When you embed an analysis in a webpage, the embedded experience may not include all the features available in the full QuickSight authoring environment. Since when embedding an analysis or dashboard, QuickSight assumes a different set of permissions and capabilities for the end-user, regardless of their actual license type within the QuickSight console. This is because the embedded experience is designed to be consumed by a broader audience, and not necessarily by users with advanced authoring capabilities.

To provide the full authoring experience, including the "Build" button and other advanced features, within the embedded context, you would need to ensure that the end-users have the appropriate QuickSight Author or Author Pro licenses assigned to them.

Additionally, you may need to configure the embedding settings and permissions accordingly to allow these features in the embedded context. In this case you may consider embedding the QuickSight console to allow full authoring experience for registered users. [1]

If you are still facing the issue, can you please open a support case with AWS using the following link and we will be happy to help you. [+] https://support.console.aws.amazon.com/support/home#/case/create

Thank you!

References:-

[1] https://docs.aws.amazon.com/quicksight/latest/user/embedded-analytics.html

[2] https://aws.amazon.com/about-aws/whats-new/2020/07/amazon-quicksight-adds-embedded-authoring-namespaces-for-multi-tenancy-and-more/