The verb ألقى: إلى for "offer", على for "cast"
Hypothesis examined: the preposition would fix the meaning of the verb أَلْقَىٰ: with إِلَىٰ and an object, "offer"; with عَلَىٰ, "cast".
The hypothesis rested on real cases: « وَأَلْقَوْا۟ إِلَيْكُمُ ٱلسَّلَمَ », "and offer you peace" (4:90), and « تُلْقُونَ إِلَيْهِم بِٱلْمَوَدَّةِ », "offering them affection" (60:1).
The test on the whole corpus shows that "offer" holds only for those objects: peace (4:90, 4:91, 4:94, 16:87) and affection (60:1). Elsewhere أَلْقَىٰ إِلَىٰ keeps its basic sense of throwing, casting or conveying: Sulayman orders « فَأَلْقِهْ إِلَيْهِمْ », "cast it to them" (27:28), about a letter; a treasure is "cast down" (25:8); the Scripture is "conveyed" (28:86).
With عَلَىٰ the meaning is not uniform either: "cast" (12:93, 12:96, 73:5), but also "place" a love (20:39) or "set" a body on a throne (38:34).
It is therefore not the preposition that fixes the meaning but the object: offering peace is a fixed expression ("casting peace toward someone"), not a rule of construction.
In the text
وَأَلْقَوْا۟ إِلَيْكُمُ ٱلسَّلَمَٱذْهَب بِّكِتَٰبِى هَٰذَا فَأَلْقِهْ إِلَيْهِمْوَأَلْقَيْتُ عَلَيْكَ مَحَبَّةًۭ مِّنِّى