A follow-up to my own query, since this information does not appear to be anywhere on the Internet. tl;dr: there is indeed a standalone computer kiosk (that looked like it could print a BP) in the sterile transborder-to-international connections area, but it was roped off and didn't appear to be available unless you already had (or at least claim to have) a BP to get into that area in the first place.

So, I ended up flying DCA-EWR-YUL-DOH on UA/AC/QR, hand luggage only. UA/AC for DCA-EWR-YUL were on one reservation, QR on another. I have both US & Canadian passports, but I went with my wife who has only US (she had two reservations of her own like me, one for DCA-EWR-YUL and one to DOH). I was able to check myself in online on UA/AC from DCA-EWR-YUL with no trouble at all, I got mobile BPs and no one ever did any document checks. Wife was (as expected) unable to check in online for those flights so we headed to DCA at T-90 to ensure time to speak to an agent. We were both able to check in online for YUL-DOH just fine on QR, so we had those BPs on hand at DCA.

At DCA, wife's status as my spouse was sufficient to get her checked in for DCA-EWR-YUL with no further explanation. I asked whether "sterile transit" or "transit only" was listed as an exception and the UA agent said that all she saw was "immediate family" of a "citizen, resident, or national of Canada" or something under the Indian Act. So that was interesting to hear, but the agent didn't really have incentive to read all the fine print in TIMATIC. Whatever, check-in was quick. Flights to YUL were uneventful.

At YUL, we disembarked in the familiar transborder corridor and walked from gate 78 towards the immigration hall. I asked someone about a connection to DOH and was advised to continue forward (without going downstairs) and approach a door on the left for connections. (This is in some conflict with new looking signage that says "NEW" and to go right for connections.) Sure enough, an agent there asked about our connection and said someone would be right with us. She asked if we had onward BPs and I said yes, but nothing was checked. 5 minutes later, another agent appeared and escorted us through what appeared to be a very new, unstaffed, and underutilized immigration room. The agent asked for our inbound BP (from EWR) and even wrote down the AC flight number but then only asked whether we had onward BPs. I said yes and we then went downstairs (from within the sterile connections area). Interestingly, as we came down the escalator into a small hall with a handful of electronic immigration kiosks, I noticed a single multi-airline check-in kiosk that was off to the side but roped off... it had perhaps 13 to 16 airlines on it (notably, not Qatar... but American, Turkish, many others). So perhaps it is indeed possible to get down there, check in on the kiosk with your airline, and continue through sterile transit. In that room, I scanned both my and my wife's passports in a kiosk, got a paper "receipt," and dropped the receipt into a bin before being escorted out into the terminal, no security check required.

As I expected, at no point did it ever even appear to matter whether my "airline" (whatever that would mean) had a practice of automatically transferring checked luggage, and so forth. So I feel confident concluding that the ADM language quoted by TechnoTourist above is simply poorly drafted language meant to convey the reality that if you have checked luggage that is tagged to YUL, then you are stuck clearing customs in YUL and rechecking. But it does not mean that in other cases your ability to use sterile transit rests upon some determination of whether luggage is transferred automatically. Rather, the important thing seems to be to have your onward BP in hand (indeed, any onward international BP will do). And now that I know there is an airline check-in kiosk in the sterile connection area, I would probably at least attempt to ask an agent to let me generate a BP there (if for some reason I couldn't get my onward BP in advance) rather than entering Canada unnecessarily during Covid.

Finally, I would note that there actually is a QR "transfer desk" that has all the capabilities that I typically associate with the interstitial transfer desks that you find in the no-mans-lands at airports like LHR and SGN--except that at YUL, it's nestled within the secure area of the international terminal, so you have to find your way there to use it. That said, I would be willing to bet that, in the right circumstance, someone from that QR transfer desk could be paged to assist an onward QR passenger who had good reason to wish to transit airside.