I’m not sure we get the order of magnitude. As His people test Him and fail to follow Him, He continues to reveal His constant, active, present-being as “I AM.” The people may fail to fully realize it or appreciate it in the present moment, which could unfold in the biblical text as growing definition in how God relates to His people over time. But that is more like an aspect of growing relationships and getting to know who someone is and has been, though I may not fully realize it now.
You brought up many interesting passages referencing God’s name being used sometimes alone and sometimes in conjunction with an angelic presence. I tried to go through them all below, fetching some Hebrew and research data on context and meaning, but please pardon the fact that I’m no expert in Hebrew and have to stand on the shoulders of those who do. As such, I also left the right-to-left word order for each below…
- Yahweh malak
- of-LORD the-angel
- מַלְאַ֧ךְ יְהוָ֛ה
“The angel of the LORD” (מַלְאַ֧ךְ יְהוָ֛ה) is mentioned fifty-eight times in the OT, “the angel of God” eleven times. Angels of the LORD appear either singly as here or in groups. When first seen, they are usually taken to be men, but by the end of the encounter one of them is realized to be God (18: 2, 22; Judg 6: 11–22; 13: 3–22). When, as here, the text simply speaks of a single angel of the LORD, this must be understood as God himself appearing in human form, nearly always to bring good news or salvation. The angel of the LORD appears frequently in Genesis and in the Book of Judges but rarely in the literature dealing with later periods. The exact relationship between the angel and God himself has been the subject of much inconclusive discussion. The Fathers identified him with the Logos. Modern scholarship has seen the angel as a creature who represents God, as a hypostasis of God, as God himself, or as some external power of God. (For further discussion, see THWAT 2: 900–908; Westermann, 2: 289–91; EM 4: 975–90; G. von Rad, OT Theology, 1: 285–89.) Within Genesis, the angel of the LORD tends to appear at moments of dire personal crisis (cf. 21: 17; 22: 11, 15).”
With the suggestion of hypostasis or the Logos quoted above as a part of biblical interpretation tradition in this passage, we could read into the text this meaning. However, there is not enough to concretely read Christ out of the text at its face value, so it’s likely an open discussion.
- Yahweh elaw wayera
- the-LORD to-him appeared
- וַיֵּרָ֤א אֵלָיו֙ יְהוָ֔ה
The identity of these visitors are not immediately apparent to Abraham, and he thought they were men (v.2). As he talks to them, he begins to understand who they are. He later addresses the LORD as Adonay (אֲדֹנָ֔י) or “sovereign,” in intercessory prayer (confer 18:3, 27, 30-32). There are different interpretations of this, like “my Lord” or “Lord of all, sovereign”, and the potential for it being plural. But the best likely representation of this formal address by Abraham is “sovereign.” 
- elaw Yahweh malak wayera
- to-him of-the-LORD the-angel appeared
- וַ֠יֵּרָא מַלְאַ֨ךְ יְהֹוָ֥ה אֵלָ֛יו
The messenger of Yahweh, is not an “angel” in the sense in which “angel” is now generally understood. As often in the OT (Gen 18, Judg 6), there is in this passage a fluid interchange between symbol, representative, and God himself. In the composite form of the present text, Moses sees the symbol (“ a blaze of fire”) and hears Yahweh (vv 4–6, 7–10, 12). Only we are told that Yahweh’s messenger appeared to him. For the redactor, there was no inconsistency: the addition of Elohim (v 4) to the messenger, the fire, and Yahweh of v 2 simply provided four designations of the same and single reality." 
It seems we can take at face value that these presences are representative of God in the truest possible sense, as far as He can present Himself before a human.
- Yahweh wayyar
- the-LORD so-when-saw
- וַיַּ֥רְא יְהוָ֖הi
Same context for the LORD as used in 3:2.
- ehyeh aser ehyeh
- I-AM who I-AM
- אֶֽהְיֶ֖ה אֲשֶׁ֣ר אֶֽהְיֶ֑ה
God’s reply is in an active state of being, not past or future, but He always is and continues to be so. This is echoed when God repeatedly affirms who He is (vv.12-14).
- Yahweh wayyered
- the-LORD and-descended
- וַיֵּ֤רֶד יְהוָה֙
- wayyaabor Yahweh
- and-passed the-LORD
- וַיַּעֲבֹ֨ר יְהוָ֥ה
God is again declaring who He is, as He did earlier in 3:16. There is a more interesting aspect in the double-calling of His own name in the same verse 6:
- wehannun rahum el Yahweh, Yahweh
- and-gracious merciful God the-LORD, the-LORD
- יְהוָ֣ה יְהוָ֔ה אֵ֥ל רַח֖וּם וְחַנּ֑וּן
There is no mistake or casual redundancy in the text as God is emphasizing the present reality of Yahweh as being, active, and present.
 Gordon J. Wenham, Gensis 16-50 in Word Biblical Commentary, vol. 2, eds. David A. Hubbard, et al. (Grand Rapids: Zondervan, 2000), Kindle edition, 8-9.
 - - , Genesis 1-15 in Word Biblical Commenary, vol. 1, eds. David A. Hubbard, et al. (Grand Rapids: Zondervan, 1987), Kindle edition, 327.
 John I. Durham, Exodus in Word Biblical Commentary, vol. 3, eds. David A. Hubbard, et al. (Grand Rapids: Zondervan, 1987), Kindle edition, 30.
 Ibid., 38-39.
 Ibid., 452-453.