OP asks whether šeqeṣ ("abominable") in Leviticus 7:21 should be taken as
"disgusting and repulsive" or in some sense "spiritually horrible" or "idolatrous"?
I think it's probably none of these. The word šeqeṣ is consistently used in the realm of ritual purity:
All eleven attestations of the [noun] שֶׁקֶץ connote detestable animals and other creatures that Yahweh designated as unclean... 1
This is clearly the case in Lev. 7:21 which concerns the state in which one must partake of the peace offering.
[OP]: [I]n other words, is Leviticus suggesting that pigs and rats are naturally intrinsically unsuitable as food...
Neither pigs nor rats are mentioned in this verse, and there's nothing here about eating them.
Which animals are they, and what is forbidden?
The phrase šeqeṣ ṭāmēʾ (KJV: "abominable unclean thing") in Lev. 7:21 is problematic not because its connection with ritual purity is in question, but because it is apparently a tautology: both words designate ritually unclean animals. Jacob Milgrom has argued that ṭāmēʾ animals may be neither eaten nor touched, whereas most šeqeṣ animals convey impurity only through eating, a pattern he discerns in Leviticus 11 (vv. 10–13, 20, 23, 41).2 Most seqeṣ animals, then, are not ṭāmēʾ since they defile only by ingestion.3
Using Milgrom's scheme, the difficulty with Lev 7:21 is not one of tautology but of contradiction, since the two groups are largely mutually exclusive. He resolves this by pointing out that Leviticus 7:21 concerns defilement incurred by touching (nāgaʿ). A subset of seqeṣ is then identified which overlaps with ṭāmēʾ:
The eating of forbidden animals is not defiling, while touching is defiling only if the carcass is that of a quadruped (11:8, 24–28) or one of the named eight nonquadrupeds (11:29–31). Hence the term šeqeṣ ṭāmēʾ [in Leviticus 7:21]....must refer to the eight nonquadrupeds singled out in 11:29–31, whose contact is defiling.2
This list includes mice, rats, and a variety of lizard-like animals. No pigs here (ṭāmēʾ only).
Notably, BHS suggests the variant šereṣ ("swarming creature") for šeqeṣ, with the Samaritan Pentateuch and the Peshitta (cf. Lev 5:2), a reading apparently accepted by the NIV. If correct, this would remove any reference to "abominable" such as concerns the OP. Milgrom argues that this variant does not change the identification of še?eṣ ṭāmēʾ. Certainly by either reading Lev 7:21 remains focused on the ritual state of the one who would eat of the offering, without mention of consumption of the še?eṣ animal and with little semblance to the modern English connotation of "disgusting".3
1. Michael A. Grisanti, "שָׁקַץ (šāqaṣ)", in New International Dictionary of Old Testament Theology and Exegesis, Ed. Willem A. VanGemeren, 4:243ff.
2. Jacob Milgrom, Leviticus 1–16, The Anchor Yale Bible; (YUP, 1974) pp. 425, 655-656.
3. There is another, more common Hebrew term frequently translated "abomination" -- tōʿēbāh -- with a somewhat broader semantic range: "something offensive to a person’s values, culture, or religion" (NIDOTTE, 4:314). It may be for this reason that "detestable" is preferred for šeqeṣ in modern translations.