My understanding is that you need to demonstrate that you are a US citizen. The requirement as that you prove that you are a US citizen to the satisfaction of the border agent, not that you have ID proving you're a US citizen. Obviously simply loudly saying I'm a US citizen alone won't do that, and having the proper ID is by far the easiest and legally unambiguous way to do that, and I would never recommend intentionally traveling to the US without a passport or other WHTI-approved document, and I've never tried it, and it would take much longer without proper ID to prove your citizenship, and you won't be able to board a flight or other commercial vehicle traveling to the US without valid ID. But under international law you have the right to enter your country of citizenship if you physically arrive at the border, as you can do walking or driving to a land border. And relevant to this thread, my point was simply that a Global Entry card which describes the holder as a US citizen proves US citizenship, and that's really all you need to enter the US. Unlike a NEXUS card, a Global Entry card declaring US citizenship is not, in my experience or by my reading of the CBSA web site, valid for entry to Canada. (I haven't read any relevant case law.) And also unlike a NEXUS card, a Global Entry card is not valid for international air travel under any circumstance.

As I recall my c. 2015 experience driving to Canada with a Global Entry card for a short visit as a non-resident, the CBSA agent actually did let me in but said the document isn't legally valid for travel; other trusted traveler cards such as NEXUS are, but they'd let me in this time. (I have all sorts of privilege; I have to assume I wouldn't have been so lucky if I weren't white, English/French-speaking, etc.) When I became a permanent resident years later at the same border crossing, the agent went into a fair bit of detail how as a permanent resident I should always travel with my permanent resident card but if I lose it or am in trouble (he specifically said "I know you're from a generally safe country, but you never know"), I still have an absolute right to enter Canada; I'll just have to satisfy the border guard that I am in fact a permanent resident. I have never tested that either, and never plan to.

But again, obviously no one should plan to ever approach any border without the appropriate ID!