Health checks fail when using the new publish to elastic beanstalk interface

### Describe the bug

See https://github.com/aws/aws-toolkit-visual-studio/issues/311. Creating a separate issue on this repo as the root cause is likely in the AWS Deploy Tool for .NET CLI which drives the new Visual Studio experience.

We should review differences in packaging between the legacy deployment experience and the new one and how that interacts with health checks.

Start with the linked Visual Studio issue 311 and trace how the AWS Deploy Tool for .NET CLI packages applications for the legacy and new publish experiences. Compare those packaging paths and verify that Elastic Beanstalk health checks succeed with the new interface; the issue is done when the packaging difference causing the failure is identified and corrected.

Written by the indexing model from the issue text.