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. 20:3838And I will purge out from among you the rebels, and them that transgress against me: I will bring them forth out of the country where they sojourn, and they shall not enter into the land of Israel: and ye shall know that I am the Lord. (Ezekiel 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 19:44It may be the Lord thy God will hear all the words of Rabshakeh, whom the king of Assyria his master hath sent to reproach the living God; and will reprove the words which the Lord thy God hath heard: wherefore lift up thy prayer for the remnant that are 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 21:13,1413And I will stretch over Jerusalem the line of Samaria, and the plummet of the house of Ahab: and I will wipe Jerusalem as a man wipeth a dish, wiping it, and turning it upside down. 14And I will forsake the remnant of mine inheritance, and deliver them into the hand of their enemies; and they shall become a prey and a spoil to all their enemies; (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 9:1414Should we again break thy commandments, and join in affinity with the people of these abominations? wouldest not thou be angry with us till thou hadst consumed us, so that there should be no remnant nor escaping? (Ezra 9:14)). And Haggai and Zechariah speaking by the Spirit of God, acknowledged them as such (Hag. 1:12-14; 2:212Then Zerubbabel the son of Shealtiel, and Joshua the son of Josedech, the high priest, with all the remnant of the people, obeyed the voice of the Lord their God, and the words of Haggai the prophet, as the Lord their God had sent him, and the people did fear before the Lord. 13Then spake Haggai the Lord's messenger in the Lord's message unto the people, saying, I am with you, saith the Lord. 14And the Lord stirred up the spirit of Zerubbabel the son of Shealtiel, governor of Judah, and the spirit of Joshua the son of Josedech, the high priest, and the spirit of all the remnant of the people; and they came and did work in the house of the Lord of hosts, their God, (Haggai 1:12‑14)
2Speak now to Zerubbabel the son of Shealtiel, governor of Judah, and to Joshua the son of Josedech, the high priest, and to the residue of the people, saying, (Haggai 2:2); Zech. 8:66Thus saith the Lord of hosts; If it be marvellous in the eyes of the remnant of this people in these days, should it also be marvellous in mine eyes? saith the Lord of hosts. (Zechariah 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. 13:99And I will bring the third part through the fire, and will refine them as silver is refined, and will try them as gold is tried: they shall call on my name, and I will hear them: I will say, It is my people: and they shall say, The Lord is my God. (Zechariah 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. 11:55Even so then at this present time also there is a remnant according to the election of grace. (Romans 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. 2:2424But unto you I say, and unto the rest in Thyatira, as many as have not this doctrine, and which have not known the depths of Satan, as they speak; I will put upon you none other burden. (Revelation 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. 3:1515But if I tarry long, that thou mayest know how thou oughtest to behave thyself in the house of God, which is the church of the living God, the pillar and ground of the truth. (1 Timothy 3:15).