Is it false to say that the Israelites remained faithful to their religion throughout all of their troubles?

Answer 1

It is false to say that ALL of them remained faithful. But there were always some who did.

Shortly before he died, Moses warned the people that he suspected that they would eventually succumb to the lure of the idols (Deuteronomy 29:17). Joshua gave a similar warning (Joshua ch. 24).
These warnings came true. Many of the Israelites went astray after the foreign gods (Judges 2:11). However, they never invented their own idol. It was always the baneful influence of other peoples. And there were times when the entire Israelite nation repented (Judges 2:1-4) and prayed to God (Judges 3:9, 3:15, 6:6, 10:10). Those who did sin did not represent normative Judaism. They were deviating from the Torah's standard; they were publicly, repeatedly, and scathingly excoriated by the Prophets, and they caused God's retribution to come upon the entire people.

Because of the idol-worship that did happen, ancient images of idols have been found in Israel too. Images of God aren't found because it is forbidden to represent Him through imagery (Deuteronomy 4:15-16).

It should be noted that idolatry was never universalamong the Israelites. The tradition of the One God was handed down in every generation, whether by the few or the many; and it is those who handed down that tradition whose beliefs we Jews continue today.

Deborah ascribed victory to God (Judges 4:14), Gideon tore down the idolatrous altar (Judges 6:25-27); Samson prayed to God (Judges 16:28), as did Hannah (1 Samuel 1:11) and Samuel (ibid. 12:18); Eli blessed in the name of God (1 Samuel 2:20), Saul built an altar to God (1 Samuel 14:35); Jonathan ascribed victory to God (1 Samuel 14:12), as did David (1 Samuel 17:46); and Solomon built the Temple for God (1 Kings 8:20). A number of the kings "did what was right in God's eyes": David (1 Kings 15:5), Solomon (see 1 Kings 3:3), Asa (1 Kings 15:11), Yehoshaphat (1 Kings 22:43), Yehu (2 Kings 10:30), Yehoash (2 Kings 12:3), Amatziah (2 Kings 14:3), Azariah (2 Kings 15:3), Yotam (2 Kings 15:34), Hizkiah (2 Kings 18:3), and Josiah (2 Kings 22:2). Part of this righteousness was their destroying whatever idolatrous incursions had occurred among the populace (1 Samuel 7:3-4, 2 Chronicles 15:8, 2 Chronicles 17:6, 2 Kings 18:4).

Even at the height of the unfortunate spread of idolatry among the less-loyal Ten Tribes, there were thousands who remained loyal to God (1 Kings 19:18).
And, of course, the Prophets, who spoke in the name of God and warned against idolatry: Isaiah, Jeremiah, Ezekiel, Hosea and so on.
The sages of the Talmud, who ridiculed idolatry (Megillah 25b), were simply continuing in the tradition of the Prophets whose verses are quoted in that context (ibid.).

See also:

Jewish history timeline

Are Hebrews, Israelites and Jews the same people?

Answer 2

While the Israelites faced periods of faithfulness to their religion, they also experienced times of straying from it. Throughout their history, the Israelites struggled to maintain their commitment to worshiping only Yahweh and often turned to idolatry and other gods.