My iPhone SE displays "no internet connection" in spite of showing that I am connected to my home network.

Mason5116 wrote:

Thank you. Killing my VPN cleared the problem.

Great, glad to hear it. VPN can be pretty persnickety. Here is some information on how to deal with VPN:

VPN frequently creates more problems than it solves. But one important fact is that when you install VPN it configures itself to the specific hardware, iOS version and network that the phone uses. If any of those change it can break the VPN. So any time you change any of these you may need to delete the VPN app and profile, restart your phone then add them back (if you still want to use VPN) so it can configure itself to the new environment.

Here's Apple's advice on VPN→

Check VPN (Virtual Private Network) and third-party software to help resolve network connectivity issues - Apple Support

And something to think about→Don't use VPN - GITHUB

There are two legitimate purposes for using VPN:

• To allow access to a private network such as a school or business when you are not on site. 
• To allow access outside of a country with a repressive government that has restricted Internet access. (This has suddenly become more important)

Any other use is risky, and can lead to problems like the one you experienced. VPN disguises your location by making you appear to be somewhere else in the world. But you usually can’t control that “somewhere else”, and if it is in a location that an app isn’t approved for the app won’t work. Plus the fact that the provider of the VPN knows everything about you and your location, as well as what sites you access through the VPN. So you are totally dependent on the VPN provider’s honesty. As a start, if the VPN is free, DON’T USE IT. The provider has to make money somehow, and if you aren’t paying them then they are selling your private data to make money. But even those that charge can’t necessarily be trusted. For example, a few years ago Avast was caught selling user browsing data. They claim they have stopped doing so. 

You don’t really need VPN when using public Wi-Fi, because all communications between your device and the servers it accesses are end-to-end encrypted.

If you want VPN for privacy about the sites you visit, that’s not a good choice as discussed; instead you should download and use the TOR browser. Or with iOS 15.2 and later for iOS/iPadOS and MacOS Monterey 12.2 and later Apple now has iCloud+ Private Relay, which is not VPN, but provides a safer browsing environment than VPN, and it doesn't spy on you→About iCloud Private Relay - Apple Support

The other use for VPN is to “steal” content that is not available in your area. I leave this question to your personal ethics.

That message means that your router is not connected to the Internet, but your phone is connected to your router.

Are you using VPN? If you are, delete the VPN app and VPN profile.

If you are not using VPN see if other devices connected to your network can connect to the Internet. If not, check with the ISP that provides your internet service.

If other devices work restart your router by turning it off for 15 -30 seconds.