The benefit of managed packages is that they can be upgraded. You can add new content to them, and those changes can be applied with a simple update. The downside to managed packages is that they have restrictions on deleting metadata from them, so it becomes very important to design your application beforehand, lest you end up with lots of stuff you can't delete. Some care also needs to be taken to not accidentally link the package to another package or unwanted metadata for the same reason. It's nearly impossible to fix these mistakes.
Unmanaged packages can be freely modified, even after installation (e.g. someone an install your package, then delete a field). The downside is that you cannot upgrade those packages. They must be fully uninstalled and reinstalled to apply a new version. This is problematic in many cases, as you need to make sure that you have all the metadata you need in one shot, and making changes is easy, but non-trivial to fix in orgs where the package is installed.
You may want to look at Unlocked Packages. They have the editability of unmanaged packages, and the upgradeability of managed packages. This would require moving to DX, which is not terribly difficult, but will take a little bit of time.
Everything is migrating to Lightning, so I'm not surprised that clicking on the object name takes you to Object Manager. The "zero objects" bit may just a bug of some sort. There's clearly one object in the package.
