It would be unusual to abandon the children to the state, but not unheard of, unfortunately. If there's a remaining parent then it seems that some sponsorship, at some point, should have been available. But whatever, the devil is always in the details.
Not taking the children with you when being removed from a nation where one did not have a legal right to remain could certainly be seen in the same light. Having children while living in a country without status isn't a laudable achievement and is so manifestly irresponsible that some might question the parents' competency to raise them responsibly.
I don't know anything of the circumstances of this particular H&C but generally speaking, if an H&C has been submitted for about a year without a decision then the CBSA is usually amenable to voluntarily suspending the removal in order to await the decision.
The Federal Court can be accessed to consider a temporary Stay of removal for an H&C decision, arguing irreparable harm and perhaps have success.
Perhaps this person has a "clear as water" positive decision awaiting them, if they've been victimized somehow for example. On the other hand if someone came on a TRV and didn't bother to leave and just decided to go ahead have some kids without any certainty of their own future - this sort of recklessness used to be referred to as being the "architect of one's own misfortune."
Most people, but not all, take their kids with them when being removed. I personally (for what it's worth) don't have any reservations about lawfully deporting people who have children, Canadian or otherwise. However, if there are positive H&C factors that will lead to PR status, I don't have any issues with that either.
Within the time frame of the visitation, the mother had every intention to leave Canada with her husband to have her child; however, due to a two week overstay of his ESTA visa on a previous visit, the father was denied entry. Rather than have the mother give birth to the child alone without the support of her husband, the decision was made to stay together in Canada for the well being of the mother and child.
If the non resident mother is removed, then the circumstances are such that the children either stay with the father in Canada without their mother, or leave with the mother and be without their father. No other options.
The children, since their birth, have been with their father and mother exclusively 24 hours a day.
The extent of harm for children separated from their parents is well documented and currently considered to be a global crisis.
Hirokazu Yoshikawa, a developmental psychologist at New York University codirects NYU’s Global TIES for Children and has done extensive research
(as well as thousands of other psychological studies) on the irreparable harm - including cognitive, social-emotional and other mental health problems for children separated from their parent.
To have no reservations to remove a parent knowing it would cause irreparable damage for the children is, in my opinion, with all due respect, abominable inhumanity regardless of government law.
You mention that a Federal temporary stay of removal is possible relative to an H&C decision. Do you know if a stay of removal is applicable to the three types of removal orders
(- departure order, exclusion order, and deportation order) prior to or during the process of an H&C application?