The construct translation of Genesis 1:1-3 does contain a temporal marker (when God began to create). But further, you said that Hebrew only places background inside a timeframe when the text explicitly gives a temporal marker (as in Genesis 2:4–7 with “in the day…”). But that simply isn’t true.
In addition to Genesis 2:4–7 and Exodus 3:1–2, there are clear examples where background clauses exist concurrently with the start of the wayyiqtol sequence without any explicit temporal marker.
Examples:
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Judges 6:11–12: Gideon is beating out wheat (background), and then the angel appears and speaks (wayyiqtol). The background and main action are obviously concurrent.
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1 Samuel 3:2–4: Eli and Samuel are lying down, the lamp has not gone out (background), and then the LORD calls Samuel (wayyiqtol). The background describes the very moment the action begins.
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1 Kings 19:9–11: Elijah comes to a cave and lodges there (background), and then the word of the LORD comes to him (wayyiqtol). The scene is continuous; no separate timeframe is introduced.
In each case, background immediately sets the scene for the main action. No temporal marker is needed to show that the background occurs within the timeframe of the narrative. That’s simply how Hebrew narrative works.
By the same principle, Genesis 1:1–2 can describe the conditions of the world at the moment God begins creating in verse 3 (which is on Day 1).