Re “verify the new browser”: yep it’s happened to me multiple times, too. Although if I’m not mistaken, the occurrence seems to be site-specific? 
Alas for me it’s usually some public utilities
Interestingly I used to see this work fine, and for some reason a few months ago it stopped. 2FA messages come through fine on my phone, but won’t come through on the Mac via Messages.
Your phone must be configured to forward SMS to/from your Mac. And both phone and Mac must be signed in to the same Apple ID.
I experience the same thing. I’ve always assumed it was part of Apple’s “Intelligent Tracking Prevention” deleting local storage databases every seven days (if you haven’t visited the specific web site during that time). This began with the release of Safari 13.1 in early 2020. One article describing the issue and its unintended consequences is from IT News:
Later releases of Safari may have expanded on these privacy protections affecting even more site’s “memory detention” data.
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Yup, it’s set up for getting SMS. All my other messages get through fine, it’s only the 2FA messages that don’t.
Strange indeed. It seems as if Apple is explicitly looking for and filtering out those messages. Maybe a security feature (so someone with access to your Mac can’t access the 2FA codes without also having your phone)?
I wonder if this situation explains some of the blocks you experienced.
Is that true?
I thought Apple was able to terminate your incoming SMS messages and relay them into their iMessage network, in which case you could receive them over WiFi. But I should test this.
I’ve had enough experience camping to see this behavior. Small messages like sms sometimes could succeed where mms or other things could not.
Yes – Apple doesn’t run SMS relays for this feature. Your iPhone is the SMS relay, and if it can’t receive the SMS, then the text won’t appear on your Mac.
Thinking about how it could work, I can’t even see how Apple could run SMS relays for this feature. They would somehow need to be able to take control of the routing for any mobile number on any network in the world, without agreement or a relationship with the carrier.
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It’s just software. There might be something in the iMessage EULA where you give Apple relay control over your SMS and then they take care of configuring that hop with Neustar or whomever and all the carriers dip that db when figuring out how to route such messages.
I should probably do a test to convince myself 
It is.
Your Mac doesn’t have any cellular capablities. It’s all provided by your iPhone. Apple certainly cannot inject themselves into your carrier’s SMS routing. Well perhaps they could in theory, but then they’d have to negotiate a deal with each and every cell provider worldwide (and what would their incentive be? why let a potential competitor inject themselves into your network?) and we’d certainly have heard about that.
No, it all happens app-side when they combine SMS and iMessage in Messages so they appear as if they came through a single unified conduit. Case in point, those SMS you receive send have a green bubble. You only get blue for actual iMessages. And gray background for this new corporate thing Apple has where they basically give companies a conduit to communicate with people through the iMessage network, but that’s unrelated to SMS. Again, just Messages making it look like they’re the same thing (apart from bubble color of course).
That SMS come through even when you see essentially zero data b/w or no voice coverage, is a feature of GSM (not technically related but similar is that you at times will succeed in placing a 911 call even when you otherwise can’t get or place any calls), it has nothing to do with receiving them on an iPhone or a connected Mac (via relay).
But don’t take my word for it, just do the experiment. Turn your iPhone off. Then have somebody on an Android send you an SMS. I guarantee your Mac won’t get it until you turn your iPhone back on. And when you reply to that it will be as a green bubble.
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I hate to be pedantic, but received messages are always light gray bubbles. It’s your sent messages that are either blue (if delivered using iMessage) or green (if delivered using SMS/MMS).
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No no, you’re absolutely right. Color only on outgoing. Fixed. Thanks. 
I sort of blame Craig Federighi for this confusion. In one Apple keynote, he talked about “green bubble friends” and “blue bubble friends,” and since then, I’ve had to remind myself that the other person or people are gray, whereas you’re green or blue. It makes more sense to me that the color should clarify a fact about the other person, not you.
When I need to get a password / log in to someone I first message them a “Hello this is me is this you?” The color of the buble tells me if I can securely send the information via messaging.
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Two points:
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You don’t need to send a message to see how it will be sent. When you start typing a name, the drop-down list will color the matches in green or blue depending on whether sending to that address will be via SMS or iMessage.
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What works for one message may not work for the next. If the recipient is having network problems, your phone may fail-over from iMessage to SMS, even if there have been previous SMS messages.
I used to see this all the time with my daughter, back when we had metered data plans on our phones (and she would therefore turn off cellular data most of the time to avoid hitting her cap). Depending on whether she was on a Wi-Fi network or not would determine how my messages got sent.
Of course, if you’re sending the test message immediately before the real message (and you wait for the “Delivered” status on the test message), then it will probably work OK.