You are still a Canadian PR, no matter what, since PR status doesn't expire in normal circumstances. But without a physical PR card, you need a PRTD to return to Canada. Apply for it only while abroad.
https://www.canada.ca/en/immigration-refugees-citizenship/services/application/application-forms-guides/application-permanent-resident-travel-document.html
U.S. citizens who are Canadian PRs are allowed to board flights to Canada without presenting a PR card or a PR Travel Document.
Any Canadian PR who arrives at a Port-of-Entry (regardless of means of transportation) will be allowed entry upon showing sufficient documentation or evidence to establish their identity and PR status, no particular or specific documents required. A passport (from home country) will generally suffice but this may be only after a wait in Secondary while CBSA verifies the individual's identity and status. Typically goes easier if the PR has some documentation related to PR status, such as a copy of the CoPR or an expired PR card. This is mostly about travelers arriving at a U.S./Canada border crossing since not many PRs travel to Canada via private jets (other than U.S. citizens, and some other particular exceptions, a PR needs a valid PR card or PR TD to board a commercial flight to Canada).
Once they arrive at a PoE, PR-Travelers carrying a visa exempt passport, especially U.S. citizens, may be waived through without any questions asked regarding their PR status. This was actually quite common prior to additional screening related to Covid. That does not mean the border officials are unaware of the traveler's PR status. Ordinarily they will be aware of it once the traveler's Travel Document is scanned into the system.
Meanwhile, currently, and noting that going forward use of the ArriveCAN app may become a permanent part of the border entry process, the ArriveCAN app requires the traveler use the same travel document upon arrival at the PoE as used in completing the ArriveCAN app procedure, and it requires disclosure of the purpose of the trip, including a PR returning to Canada. Misrepresentations made in such transactions are serious, so PRs would be prudent to honestly disclose they are a returning PR.
Note though that I think for most PRs it would be advantageous to present themselves as PRs - because increasingly entries and exits are going to be recorded and hence base evidence supporting their compliance with the RO.
The question may be practically moot given the disclosures that must be made in completing the ArriveCAN app, which is for Covid-related screening but which requires the traveler to use the same Travel Document in the app as they will present at the PoE upon arrival, and requires significantly more information than typically asked of U.S. travelers arriving in Canada prior to covid. Including if the traveler is a PR returning to Canada. While it is getting a good deal of pushback, it appears that ArriveCAN will be revised for long term use, and will undoubtedly facilitate digital screening that is likely to identify Canadians as Canadians regardless of the Travel Document they are using with the ArriveCAN app.
Apart from that, generally the PoE official will see that a Canadian PR presenting a U.S. passport is a Canadian PR, and even if the PIL officer takes no note of that, the entry will almost always (with some exceptions) be recorded to the PR's CBSA travel history. Even twenty years ago U.S. travelers into Canada had a CIC (now IRCC) client number and if the entry was scanned into the system, there was a travel history record of it (just that it was around that time, following increased border controls because of the 9/11 attacks in the U.S., that border officials were beginning to regularly check and scan travel documents for U.S. citizens traveling into Canada . . . prior to that I may have entered Canada well over a hundred times without showing any identification at all, not so much as stating my name).
That is, generally there is no need to present oneself as a Canadian PR for the entry to be collected and recorded in a "client's" CBSA travel history. The system connects bio-data and client number (and if it doesn't, that is likely to trigger a referral to Secondary, where further processing will almost certainly make the connection).
I will have to check, believe I have a reference somewhere to effect that a PR or citizen must identify themselves as such at border. If I'm mistaken, then mea culpa. (I know it is a requirement by law in USA and thought I had seen such for Canada)
Re U.S. Rules:
I do not know U.S. law governing Green Card holders.
When entering the U.S., travelers are almost always asked, in an explicit question regarding each one individually, what their nationality is. If the traveler is a U.S. citizen, the U.S. does not recognize any other citizenship or nationality, so the traveler is obligated to declare their U.S. citizenship. However, I do not know the penalty for declaring another citizenship without disclosing U.S. citizenship. A hassle almost for sure. How much so I do not know. Even if the traveler says, in response to the question about nationality, "I am a U.S. and a Canadian citizen," that can trigger the side of American officials I rather dislike, depending on how anal the officer is (and it seems many are rather so) . . . since, it is not possible, pursuant to American law, to be both a U.S. citizen and a Canadian citizen under their law (dual citizenship, meaning having citizenship in another country in addition to the U.S., is not prohibited in U.S. law, but the citizenship in another country is not legally recognized by U.S. authorities).
Canadian Rules:
Not aware of any statute, regulation, or administrative rule that requires Canadians to specifically identify their Canadian status during a border examination, so long as the traveler otherwise positively and truthfully identifies who they are and presents proper documents when requested.
If asked, of course a PR must be truthful. I have seen a number of IAD decisions including discussions about PRs who have made overt misrepresentations during a border crossing, and now (since 2015 amendments as I recall, or maybe those were in 2012) this is grounds for deeming the individual inadmissible. So it can be serious.
Again, this may be moot going forward, with the expansion of the ArriveCAN app, if it gets implemented for the long term. Since, if asked a question for which a failure to disclose PR status would be misrepresentation, of course the PR must truthfully disclose their Canadian status. Whether the current ArriveCAN app questions (such as requiring an honest statement of the purpose of the trip) effectively make it a misrepresentation to not disclose the traveler is a returning Canadian PR, I am not sure. Not something I'd recommend trying, especially since even without disclosing one's PR status, screening officials can easily see when a traveler is a Canadian PR (with exceptions, of course, as always), so the odds are not disclosing PR status does not hide that the traveler is a PR.
