Depends on the distro. Many distros don't integrate perfectly, some require additional configuration/setup around PackageKit, and there are a lot of customer complaints around various basic issues. One common example: the software centre showing "No Application Data Found". Another common issue: updates don't really work, the workaround is to use the underlying package manager to perform updates.

GNOME's software centre issues very clearly stem from the diversity of package managers and formats and environments it has to support because Linux actually has a decentralized application ecosystem. On Android, F-Droid likes to claim that the "technical skill" needed to install other packages is an artificial barrier which is just imposed by Google, yet it merely piggybacks off the usability & simplicity of Google's centrally developed package formats (.apk or .appbundle) and package manager (PackageManager). When the ecosystem is actually decentralized, GNOME software shows just how user unfriendly things become.

Yeah, as far as I've tried to use it, I've found it's pretty good on Fedora (basically the flagship GNOME distro) and that's mostly it. It hasn't worked very well at all for me on Ubuntu.

What's a package manager? Is that an app store?

gnome's software center is fine if you know what you want and it's in the repos. the discoverability is essentially nil though - yeah, there's some browsing and categories, but the categorization is poor and more popular apps aren't surfaced in any meaningful way.

it's great if you're an indie software developer who cares about things like fairness and other apps not being featured above your own, but it's certainly a long ways off from the app store experience of android or iOS in nearly every metric that my mum might care about.