I am creating an unlocked package that currently only contains custom objects, fields, and other related metadata. I am also using a permission set to give a user access to this metadata. However, I found that after packaging, creating a package version, promoting the version, and installing it in an org, the System Admin profile seems to have full access to all the members of the unlocked package in the installation org. The package itself does not contain any profiles, and neither does the SFDX project that I am working in. I am not sure how or why this is happening, so any input/help with this is appreciated!
From the unlocked package documentation:
When you install a package with the CLI, all users are given full access to all the metadata contained in the package. We acknowledge that this experience is not optimal, and plan on improving it in the future.
So it's probably not just the System Administrator profile.
Are you installing by CLI or browser? But even via browser, your options will still be to install for Admins only, or all users. I don't think there's a way to install a package so that not even the System Administrator standard profile is granted access to the contents.