Can't approve devices because approval request never appears on other devices

This doesn't help. My problem is not that I'm not able to get codes—I'm stuck on "Waiting for Approval". I'm not getting the notifications to approve the device that precede the codes.

If I sign into iCloud.com from a new browser, I get a pop-up on all three devices (Mac, iPhone, and iPad) showing a map and asking me if I want to allow this sign-in attempt from a new device. If I click or tap "Allow" on any of the three devices, the pop-up disappears, and two of the devices will show 2FA codes. (As I mentioned above, the iPad has not yet been verified/approved to receive codes.) I enter whichever code appeared last, and I'm in. I can also ask for a new code to be sent by text, and that works as well. But this is working only for iCloud on the web.

If go into Settings (or System Preferences on the Mac) on any of the three devices and click or tap "Continue" under the Update Your Settings prompt, and enter the appropriate passwords/passcodes, I get the Approval screen on the device I'm on, but that pop-up with the map asking me if I want to allow this sign-in does not appear on any of the devices. Getting a code from Settings or via text doesn't help, because until I get past the Approval screen, there's nowhere to enter a code.

I don't need account recovery. I have full access to my iCloud account via the web. What I need is to get the OS on my three devices fully signed in so that they can share iCloud services with each other. And to do that, I need to receive the approval notices that I'm not currently getting for the devices (but that I do get for web sign-ins), or I need to not need these approvals.

Given that I'm not seeing reference to these approval notices anywhere with regard to 2FA, I'm starting to suspect that they are an artifact from the older, apparently deprecated two-step authorization feature. I had been using this in the past, because where 2FA did not work for me, two-step did. But if the two don't play well together, and that's what's causing my trouble, I need to find a way to remove the two-step authorization from my account.

Update: I get an approval notification on all three devices when I try to sign into iCloud.com on the web, including the device I'm using to access the web site. The notification is properly followed by a 2FA code—but I get two different codes, one on the MBP and one on the iPhone. (The iPad isn't approved for receiving 2FA codes yet.) I have to use the one I received later to get access.

Also, I rebooted all three devices since my original post. All three are currently indicated as being signed in to iCloud, and all three are asking me to update settings. All three get stuck on waiting for approval, and no notifications to approve appear on any device. I'm getting approval notifications only when signing into iCloud on the web.

I've never gotten a real resolution. I ended up rolling back 2FA before the two-week grace period ended, just to shut the devices up about the sign-in. (When you switch to 2FA, Apple sends you an email with a link that's good for two weeks allowing you to revert to not using 2FA.) I still can't use the iCloud services that require 2FA, but I've managed to find workarounds in most cases.

If I do choose to try Apple's 2FA again, I'll activate it from the Apple ID web site rather than from within a device's settings—that was the one place I could sign in and get the correct prompts from other devices to allow me to continue. But I'm in no hurry to do so. I don't expect perfect security from my technology, but I do expect the security measures I use to work virtually 100% of the time. A security measure that won't let me properly access my accounts every time is nothing but a waste of my time.

You're welcome. I figure that the more relevant detail I provide about my problems, the easier it will be for others to determine whether any solution I find might help them.