iOS 13 Reminders on Mac won't update

As I stated, you cannot update the reminders on MacOS until macOS Catalina comes out in October. Per the link you gave me:

If you upgrade your reminders on your iPhone with iOS 13, your iPad and Mac using the same iCloud account can’t access your reminders until iPadOS and macOS 10.15 Catalina are available.

BUT:

Also, per the link you gave me, you can view the reminders from:

A web browser on a Windows PC or an Apple device, even if it's running an earlier version of iOS or macOS. Go to Reminders on iCloud.com and sign in with your Apple ID.

Thus: until macOS Catalina comes out in October, you'd have to go to the iCloud website to view your reminders on your Mac computer.

The new version of Reminders should come with the release of MacOS Catalina later in October.

No worries, glad to have helped!