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Gorman's rule is clearly not based on the etymology of the words, since historically, the alternation of s and r did in fact originate from rhotacism and not from a historical process of r-deletion. (I don't know the historical origin of the nouns like pollis that have n in the stem but not in the nominative.)

TKR pointed out in the comments that Classical Latin also has many words that lose t in the nominative that come from Proto-Indo-European i-stem nouns. Mons and sors are two examples. The nominative forms of words like these seem to originate from syncope of i, followed by simplification of the cluster [ts] to [s]. Since this syncope was a Latin-specific development, and preceded the cluster simplification, these seem to require a Latin-specific rule at some point that changed [ts] to [s] in this context.

Gorman's rule is clearly not based on the etymology of the words, since historically, the alternation of s and r did in fact originate from rhotacism and not from a historical process of r-deletion. (I don't know the historical origin of the nouns like pollis that have n in the stem but not in the nominative.)

TKR pointed out in the comments that Classical Latin also has many words that lose t in the nominative that come from Proto-Indo-European i-stem nouns. Mons and sors are two examples. The nominative forms of words like these seem to originate from syncope of i, followed by simplification of the cluster [ts] to [s]. Since this syncope was a Latin-specific development, and preceded the cluster simplification, these seem to require a Latin-specific rule at some point that changed [ts] to [s] in this context.

I found a paper that describes some theories about when this happened and what the intermediate steps were: "The Proto-Indo-European *-VTs# clusters and the formulation of Szemerényi’s Law," by Dariusz Piwowarczyk. The main example word he uses actually has "d" (*pod-s) but it seems that all scholars agree that underlying /ds/ was already devoiced to [ts] in PIE.

According to Piwowarczyk, Oswald Szemerényi thinks the a simplification of word-final [ts] to [s] in nominative forms happened in Proto-Indo-European. The processes involved are anticipatory assimilation of [t] to the following [s], with compensatory lengthening of the preceding vowel when the resulting geminate [ss] simplifies to single [s]. Examples of this in Latin would be words like pēs.

Piwowarczyk himself argues that the *Ts clusters were simplified independently in different branches of Indo-European.

I found a paper that describes some theories about when this happened and what the intermediate steps were: "The Proto-Indo-European *-VTs# clusters and the formulation of Szemerényi’s Law," by Dariusz Piwowarczyk. The main example word he uses actually has "d" (*pod-s) but it seems that all scholars agree that underlying /ds/ was already devoiced to [ts] in PIE. There is disagreement about whether this [ts] was simplified to [s] in PIE or independently in the later branches of Indo-European.

In "Latin Rhotacism for Real," Kyle Gorman describes the deletion of /t/ in the nominative form of words like mons, montis as part of a larger pattern of deletion of coronal consonants which also applies to words like pes, pedis and pollis, pollinis. The main argument of the paper is that nouns with variation between r in some forms and and s in the nominative in Classical Latin should be synchronically analyzed according to this deletion rule, rather than according to a separate rule of intervocalic rhotacization. So if Gorman is right, the deletion rule was still active in Classical Latin.

[CORONAL, +ORAL, -CONTINUANT] → ∅ / _]_σ

In order to prevent this rule from improperly deleting multiple consonants (such as the n in mons) Gorman proposes that the rule is "non-iterative" and "rightward-applying":

For nouns with with stems that end in -t- and -d-, the historical source of this variation seems to be a phonetic change that applied in conditions similar to Gorman's synchronic rule, where word-final [ts] was phonetically simplified at some point to [s]. It seems to me this could be considered a form of deaffrication, which is a common sort of change. Unfortunately, I do not know the underlying phonetic motivations for this change, but TKR's explanation about the auditory similarity of the sounds seems convincing to me.

I don't know the historical origin of the nouns like pollis that have n in the stem but not in the nominative. The nouns that have r in the stem but not in the nominative generally originated from the historical process of rhotacism described by the answers to Rhotacism: why?

A synchronic analysis: deletion of coronals before s within a syllable

In "Latin Rhotacism for Real," Kyle Gorman describes the deletion of /t/ in the nominative form of words like mons, montis as part of a larger pattern of deletion of coronal consonants which also applies to words like pes, pedis and pollis, pollinis. The main argument of the paper is that in Classical Latin, nouns with variation between r in some forms and and s in the nominative should be analyzed according to this deletion rule, rather than according to a separate rule of intervocalic rhotacization. So if Gorman is right, the deletion rule was still active in Classical Latin.

[CORONAL, +ORAL, -CONTINUANT] → ∅ / _s]_σ

Gorman extends this rule to nouns with stems ending in n such as pollis and sanguis by removing the [+ORAL] specification.

[CORONAL, -CONTINUANT] → ∅ / _s]_σ

In order to prevent this extended rule from improperly deleting multiple consonants (such as the n in mons) Gorman proposes that the rule is "non-iterative" and "rightward-applying":

This is the basis for his argument that forms like mos, moris can be analyzed using deletion of r before s, even though r is not deleted in the nominative forms of words with stems ending in rt such as ars, artis.

Etymological analysis

Gorman's rule is clearly not based on the etymology of the words, since historically, the alternation of s and r did in fact originate from rhotacism and not from a historical process of r-deletion. (I don't know the historical origin of the nouns like pollis that have n in the stem but not in the nominative.)

Nonetheless, for nouns with with stems that end in -t- and -d-, the historical source of this variation seems to be similar to Gorman's synchronic rule. In some ancestor to Latin, a phonetic change occurred where word-final [ts] was eventually simplified to [s].

Etymological consonant stem nouns

I found a paper that describes some theories about when this happened and what the intermediate steps were: "The Proto-Indo-European *-VTs# clusters and the formulation of Szemerényi’s Law," by Dariusz Piwowarczyk. The main example word he uses actually has "d" (*pod-s) but it seems that all scholars agree that underlying /ds/ was already devoiced to [ts] in PIE.

According to Piwowarczyk, Oswald Szemerényi thinks the a simplification of word-final [ts] to [s] in nominative forms happened in Proto-Indo-European. The processes involved are anticipatory assimilation of [t] to the following [s], with compensatory lengthening of the preceding vowel when the resulting geminate [ss] simplifies to single [s]. Examples of this in Latin would be words like pēs.

Piwowarczyk himself argues that the *Ts clusters were simplified independently in different branches of Indo-European.

Etymological i-stem nouns

TKR pointed out in the comments that Classical Latin also has many words that lose t in the nominative that come from Proto-Indo-European i-stem nouns. Mons and sors are two examples. The nominative forms of words like these seem to originate from syncope of i, followed by simplification of the cluster [ts] to [s]. Since this syncope was a Latin-specific development, and preceded the cluster simplification, these seem to require a Latin-specific rule at some point that changed [ts] to [s] in this context.