If “faith” is an inner faculty of regenerate persons rather than a pattern of behavior that characterizes faithful covenant members, then we should expect it to be described that way in the Old Testament as well as the New.
The NT frequently quotes Habakkuk 2:4, “Behold, his soul which is lifted up is not upright in him, but the just shall live by faith“: The LXX, so far from understanding the faith in question to be an inner faculty of the righteous man, has ὁ δὲ δίκαιος ἐκ πίστεώς μου ζήσεται — “the just man shall live from my (sc. the Lord’s) faith”, meaning that the righteous shall live because of the faithfulness of YHWH.
But what is more telling is the Hebrew, which uses באמונתו (be’amonato). This word amonah is the usual Hebrew word for faith, but it has its etymological roots in the word אמן, from which we also get the word “Amen.” That root means “firm”, and the first sense of amonah is also firmness. It occurs 49 times in the Hebrew Bible. Koehler-Baumgartner (HALOT) defines it as (1) steadfastness (e.g. Moses’ steady hands in Ex. 17:12), (2) trustworthiness, faithfulness (1 Sam. 26:23, Is. 11:5, Hos. 2:22, 2 Chr. 19:8, Hab. 2:4 etc. ). It is very frequently used as an adverb with the prefix -ב, “in faithfulness” or “faithfully.”
Significantly, amonah is an abstract noun formed from an adjective. This is the opposite of Greek, in which πιστεύω (“believe”) and πίστος (“faithful”) are derived from πίστις (“faith”). There is thus in Greek at least some plausibility to the claim that faith is a faculty, and that the verb derived from it means “use the faculty”. (I don’t think that’s actually the case, but it is possible, given the chain of derivation.) In Hebrew, however, the verb אמן is the root on which the abstract noun אמונה is formed. This isn’t decisive, of course — it would be fallacious to argue from linguistic priority to ontological priority — but it is highly suggestive.
In Dt. 32:4, the word is used of YHWH: “He is the Rock, his work is perfect: for all his ways are judgment: a God of amonah and without iniquity, just and right is He.” Here, the term denotes God’s faithfulness in fulfilling his promises (so Strong’s concordance and HALOT).
All this to say that there is zero evidence in the Old Testament that amonah is a faculty like a mouth or an eye that the regenerate use to do things. “Use your dependability” is a very odd way of talking. “Show that you are reliable” is more normal. Amonah is behavior that characterizes the righteous, whether God Himself or those who show loyalty among His people. It is not a faculty imparted by the Holy Spirit to the regenerate.