I recently stayed at an inn located in a rural Chinese village. Both Chinese and foreigners stay there.

I presented my passport as per the usual, but with a wave of the hand they indicated they didn't need it. I spoke with the owner of a different overnight establishment that caters mainly to foreigners.

Was told this failure to look at/copy passport constituted a breach.

Also that when taking lodging in China, the establishment accepts a level of responsibility for the guest, such as in the case a guest goes missing.

This could possibly explain why the hostel would fail to take passport, in order to forego that responsibility.

I was advised to insist a lodging place always make note of my passport.

Thoughts on this?

I wouldn't worry about it

If there is a breach, it is the hotel that is at fault. Most of the time I am asked for the passport, but not always.

Hotel registration in China is compulsory. For this hotels have a very easy system that simply copies the data from the ID card everybody has with an ID card reader.
Booking in foreigners is much more complicated and requires the manual input of the data. Many of the smaller places simply lack the skills to do so as many of the staff in these places is practically illiterate. Officially registration requires the scanning in of your passport into the official registration software. Many smaller places don’t have the scanning equipment or it is broken down or they don’t know how to use it.
Recently I noticed more and more problems with hotels where I had previously stayed before refusing me stating that the regulations had changed. This was in Chuxiong. Another place only there accepted me after getting police approval, for this they sent pictures of my passport and of myself by wechat to the police.
On another trip the police came to my room and asked me to join them to the police office for registration. The good thing was they offered me dinner afterwards in the police canteen. Two days earlier, in a different town, the police approached me on the street and asked me to lead them to my hotel where they checked the registration and even searched my room and luggage. An other time the police came and simply made pictures with their phone of my passport.
Later on that trip, in a different place, the hotel owner asked me not to wander outside at night in order not to attract attention.
Often in small places they don’t register you at all just to avoid all hassle and this is probably what happened in your case.
It often helps to leave a photocopy of your passport at reception so they have plenty of time to figure out how to register you or can wait till more experienced staff arrives.
Anyway there is no universal system. It changes from region to region, town to town and changes over time as well.

I will carry photocopies of my passport in the future.

On a related note, my Chinese wife registers for us at the hotel when we travel.

Sometimes they ask for my passport, sometimes not.

Lately they have been asking less.

Simplest to ignore the issue, it rarely if ever leads to any problems of note, at least for the traveler. The local person or hotel manager will be much more aware of possible problems than you will and will act accordingly and sensibly. Technically, I think you're right: legally you must be registered where you are staying, but I have successfully talked my way around such issues more than once. If you rent a flat, however, you will likely have a problem when you go to renew your visa, if you do not register at the local copshop.