Can’t connect iMac to Apple Services, App Store, and iCloud from home network

Read If you can't connect to the App Store, iTunes Store, or other Apple services - Apple Support.

How can I restore missing top icons and access purchases in App Store on High Sierra iMac? discusses a number other of possible solutions.

Your iPhone is running a much newer operating system than High Sierra. I would not be surprised if the App Store stops working for older operating systems. There had been a brief period of time it stopped working for High Sierra in the recent past but that situation corrected itself after a few days.

Have you installed and run any "cleaning", "optimizing", "speed-up", anti-virus or VPN apps on your Mac?

Try the following:

1 - shut down your modem, router and Mac.

2 - reboot each in the following order: modem, router and then your Mac.

See if that helps the problem.

If the problem persists try the following: boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.

NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.

It definitely sounds like you need to chase down a network-specific issue here, not a device or OS limitation—especially since the iMac works fine in your tech's workshop, and your iPhone connects without issue at home. That narrows things down quite a bit.

As for minimum macOS requirements: macOS High Sierra (10.13.6) is still technically capable of connecting to iCloud, the App Store, and other Apple services, but Apple has been slowly phasing out support for older OSes. You might start seeing hiccups, particularly with services that require newer security standards (TLS 1.2+). Still, outright inability to connect typically points more to network misconfiguration than software compatibility in this case.

The support article that John Galt has provided you gives you a number of things you can try. Here are a few more:

• Check the DNS Settings on your Mac: Apple services are very picky about DNS resolution. Try setting your iMac’s DNS to [Cloudflare (1.1.1.1, 1.0.0.1)] or [Google DNS (8.8.8.8, 8.8.4.4)]. By default, they are typically configured for your ISP's DNS servers.
  • Go to System Preferences > Network.
  • Select your Wi-Fi connection > click Advanced.
  • Go to the DNS tab, add the DNS servers, and remove others if needed.
  • Click OK, then Apply.

• Router/Firewall Filtering: Some routers or firewalls can block specific Apple service ports or domains. You may want to temporarily disable any filtering/firewall to test connectivity.

• Date/Time Sync: If your iMac’s clock is off, SSL certificates won’t validate and Apple services won’t load.
  • Go to System Preferences > Date & Time and ensure it’s set to update automatically from Apple’s servers.

• Wi-Fi Isolation or Captive Portals: Make sure your home Wi-Fi network doesn't have client isolation turned on (some routers do this by default), which can cause issues with Bonjour and other Apple services.

• Finally Try Ethernet: Just to rule out Wi-Fi issues completely, if you have a USB-to-Ethernet adapter, see if a wired connection gets through.

I'd try Ethernet first, honestly, just to confirm Internet works.

In addition to the rest of your list:

• WiFi Security Settings: High Sierra supports WPA2 and should support WPA2/WPA3 transitional. However, some routers will not play nicely with older macOS version that don't fully support WPA3. If your router WiFi security is set to WPA3 Personal, downgrade it to WPA2 Personal and see if the computer can connect.

You're very welcome!