Yes. There's no doubt that the H and C doesn't stop deportation. You could do the same thing with reopenings at the RAD, as Raoul said.
The reason you need to have the power for reconsideration is that there are often long delays between the decision and the removal, and things can change. You could have, for example, a regulation that says that you can apply to reopen your case at the RAD or can apply for a PRA at the RAD, and if it's been less than a year from your decision, there's no automatic stay of removal, or something like that. There are things you could do that would prevent the type of abuse the minister is concerned about, right?
In the case of an H and C, as you said, there's no automatic stay. We can go to the Federal Court and get a stay, but the chances are very remote, and it's only in extremely unusual cases that the Federal Court will decide that it's appropriate to issue a stay. For the vast majority of people, the mere fact that you apply for the H and C doesn't delay the removal.