Thank you for the comments and trying to replicate the issue. Since all my attempts failed, I reached out to AWS support and they mentioned, they refreshed my account and after which I am able to deploy the samples or the startup guide examples without any issue. This was their comment

I would like to inform you that Based on your account usage, the quotas related to Apprunner service were restricted for your AWS Account due to which Apprunner instances were not getting launched in Apprunner service due to which the application wasn't getting started (consequently not generating application logs), as a result, the healthchecks were getting failed. Also, your account activity is constantly monitored within each Region, and these quotas are automatically increased based on your usage.