Apart from the macabre . . .
No need to make much of this, but to be clear I am NOT the source for references to cessation of refugee status as "automatically" terminating PR status. And I otherwise doubt the term "automatic," as used in this context, has much significance, let alone any legal importance. Nonetheless, I understand both what it means in this context and why it is used by the courts and the RPD in reference to how the law operates in regards to cessation of refugee status resulting in loss of PR status.
While I apprehend that the extent to which we differ is likely minimal, mere minutia, for those potentially affected (which is those who obtain PR status as refugees or protected persons and who may be at risk of cessation) it is important to be fully aware of the operation of law which effectuates the loss of PR status upon cessation of protected person status.
There is NO application to terminate PR status involved. I do not know the extent to which those affected are given notice that they will lose PR status upon a final determination of cessation any more than those who apply for citizenship are given notice that they will lose PR status upon becoming a citizen. Note, for both, that is the loss of PR status upon becoming a citizen as well as upon cessation of refugee status, it is Section 46(1) IRPA which specifies that these events result in the loss of PR. The difference in consequence is of course huge, the loss of PR status having no negative impact on those who become a Canadian citizen, whereas the impact can be devastating for those who lose PR status because there has been a final determination of cessation of refugee status (depending on the particular grounds for cessation).
So, in this context, perhaps the references by the courts and RPD to the operation of law as "automatically" resulting in the loss of PR status is a matter of emphasis, or perhaps a matter of clarification, just to make it clear that proceedings in cessation will, if there is a determination of cessation, result in the loss of PR status as well as ceasing refugee status (again, depending on the grounds for cessation) .
For purposes of the OP's query, this is a largely an irrelevant tangent. The OP has already been the subject of the operation of law prescribed in Section 46(1), resulting in the OP's loss of PR status upon becoming a Canadian citizen. (There could also be a determination of cessation of the OP's refugee status due to becoming a Canadian citizen, pursuant to the Section 108(1)(c) IRPA, which is not automatic; however, since this would have no effect on their status as a Canadian citizen there is no prospect the government would commence cessation proceedings.)
In any event, again for those potentially affected, this tangent is a reminder that whether cessation results in the loss of PR status will depend on the specific grounds for the cessation determination, and how this operation of law works warrants some further discussion given the recent Federal Court decision in Wang v. Canada, 2024 FC 632, https://canlii.ca/t/k48ft which set aside a cessation determination because the RPD failed to consider whether cessation in that case should have been based on different grounds (Section 108(1)(e) rather than Section 108(1)(a) IRPA), grounds that would not result in the loss of PR status. Which is a subject to be further discussed in the topic specifically about cessation of refugee status and resulting loss of PR status: https://www.canadavisa.com/canada-immigration-discussion-board/threads/refugee-status-cessation-and-prs-applying-for-citizenship.333455/page-59
In that topic I will also quote and link some of the official and authoritative sources which refer to cessation of refugee status as "automatically" terminating PR status.