You ask about the term remnant. You are correct that it is used in. Scripture of God's earthly people Israel, but not of the Church of God.
Israel being God's people not by profession merely, but in truth, whether they were individually converted or not, the Lord in executing judgment on them for their sins did not exterminate them, but left a residue, a remnant; in the land after the Assyrian invasion of Samaria, and will yet bring back a remnant when the time for their final blessing shall arrive.
Now at no time since the capture of Samaria by Sargon, has the whole nation been in the enjoyment of divine favor; nor will it ever be, as a whole, though the twelve tribes will be preserved, and reinstated in the land, as Ezekiel states. But from him we also learn that not every individual among the ten tribes who will be recognized as of the seed of Jacob, will re-enter the land of promise; the rebels and transgressors amongst them will be purged out, whilst the tribes are on their way (Ezek. 20:38). As to the Jews restored to their land in unbelief, two parts will be cut off and die, and only the third part will be finally preserved; so it will be in the future as it has been in the past, that only a remnant of the nation will share in the favor of God.
From the days of Hezekiah the term remnant became applicable, and was used. He himself used it, asking the prophet to lift up his prayer for the remnant that was-left (2 Kings 19:4). The answer of God by Isaiah assured the king of that remnant's then preservation (v. 31). Subsequently God announced, that the remnant was to be forsaken because of their sins (2 Kings 21:13,14). That took place under Nebuchadnezzar, whose dynasty falling, and with it the Babylonish monarchy, paved the way for a remnant to return in the days of Zerubbabel and Ezra, in order that the Messiah might come by whose death Israel will by and bye be finally blessed.
Hezekiah viewed all that remained as only a remnant in his day. Ezra viewed those that returned in a similar light (Ezra 9:14). And Haggai and Zechariah speaking by the Spirit of God, acknowledged them as such (Hag. 1:12-14; 2:2; Zech. 8:6), and the latter gave hopes of blessing for the remnant in the future (8:11, 12). These blessings will surely be made good, but it will be only to a remnant, as Zech. 13:9, tells us.
Meanwhile, a remnant still shares in blessing, and becomes part of the Church. Of this Paul was a witness, and, with all the other believers from Israel makes up the remnant, according to the election of grace (Rom. 11:5). At whatever period then, of their national history from Hezekiah's days and onwards there has not been, nor will there be, anything but a remnant, which will in the future enjoy all the national blessings of which the prophets have written.
Turning 'our thoughts to the Church it is manifest the term remnant would not apply, for the assembly is viewed as existing as a whole on earth, though in one sense, as spoken of in Eph. 1,22, it is not all on earth at any one time. But in both its local and general aspects, it is regarded as on earth, and in the enjoyment of the privileges of the Church of God.
There is, perhaps, a confusion in some minds between the remnant" and" the rest" (Rev. 2:24.). Addressing the local assembly at Thyatira the Lord distinguishes between those unfaithful and those faithful, and calls these latter " the rest," thus marking them off as distinct in His thoughts from those who aided and abetted Jezebel; but in doing that, He addresses the local assembly as a whole. Much instruction we may surely derive from the remnant of, Haggai's days, but a remnant, or " the remnant " is a term one would not use with reference to the Church. We should deny thereby, the existence of the Church as a whole on earth viewed in its general aspect, about which Paul wrote in 1 Tim. 3:15.