The issue is in translation. Specifically how Shema
“Hear, O Israel: The Lord our God, the Lord is one.”
Deuteronomy 6:4
“Hear, O Israel: The יהוה our Elohim, the יהוה is echad/united.”
Deuteronomy 6:4
The word for God, singular, is El, the plural is Elohim. There is NO evidence of a royal pronoun used in antiquity much less in the Bible.
Because the word echad and hen in the LXX Koine is translated (correctly) as one the connotation of the word one in the Latin, and then all the other translations invokes the idea of singularity BUT
The text does NOT say God is singular otherwise the word yachid would have been used in the Hebrew and monos in the Koine.
As an example
“Therefore a man shall leave his father and his mother and hold fast to his wife, and they shall become one flesh. And the man and his wife were both naked and were not ashamed.”
Genesis 2:24-25
In the English one might think this means they were conjoined. But the Hebrew word is echad not yachid, they were united, they were in unity, in purpose, direction, and intentions. We know they were two separate bodies because the very next very says they were BOTH, so clearly the echad was not used to convey singularity but unity.
In the Shema it says that Elohim is echad. And Elohim is the plural of El. A singular person is not united, it cannot be, but the three persons of the Godhead are united.
Example of all three persons of the Godhead being mentioned in the same verse
“Draw near to me (God is speaking person 1), hear this: from the beginning I have not spoken in secret, from the time it came to be I have been there.” And now the Lord God (God person 2 is sending God person 1) has sent me, and his Spirit (God person 3 distinct from person 1 &2).” Isaiah 48:16
To answer the question about the golden calf
The reason the Hebrew text says Aaron told the people the name of the calf was Elohim is because they knew that the three persons of God delivered them out of Egypt. So it only made sense to call the image after the Elohim not the El who took them out of slavery.
The reason translators use God and not Gods is because in English El and Elohim are not translated with distinction, both are translated God like fish and fish or moose and moose, both singular and plural spelled the same. But if we were to be true to the Hebrew God inspired text the correct translation should be the name of the calf is Gods. And every time Elohim is used Gods should be used and when El is used then God singular should be used.
The objection to Gods is purely theological because they text and context clearly says Gods not God.
And to your last question OP, the calf was an image or an image bearer or a representation/representative of another. In this case the calf was an image of Elohim, three persons. It only made sense to name it after all three not just one. Because it was Elohim who took them out of Egypt not just El. No the calf was not sharing the throne. Idols do not share thrones, they represent another deity or deities. The calf was all the more insulting because it was just one calf.