No error should be written by the provider when a package is not found.
Add -ErrorAction Ignore to Find-PSResource calls.
Install-Package-Name Docker.DockerDesktop -PassThru -Verbose
VERBOSE: Performing the operation "Install-Package" on target "Docker.DockerDesktop".
VERBOSE: Installing 'Docker.DockerDesktop' package.
VERBOSE: Calling 'PowerShellGet' provider.
Install-Package: Package 'Docker.DockerDesktop' with requested version range '(, )' could not be found in any trusted repositories