So, a few things -
The reconstructed Proto-Semitic word 'il denotes a deity - there are numerous cognate words in ancient Semitic languages.
There is, however, another P-S word, 'ill which denotes “shining” - the two are admittedly very close. Indo-European roots/words have the same issues, by the way.
Two separate words- attested in Akkadian and written in cuneiform using different ‘letter’ forms.
Bottom line is that there are two words that probably sounded somewhat similar but were written differently and had very different meanings.
As others have mentioned - when Elohim is used in the Bible, the grammar is typically singular. I suppose you could argue that as the people that became the Jews moved towards monotheism, the word for ‘god(s)’ was already too well established; rather than changing the word itself, it was simply referred to using grammatical forms indicating the singular.
It is quite possible and actually quite likely however, that the two concepts (i.e. ‘gods’ and ‘shining ones’) are intertwined.
In many Indo-European languages, the words for ‘god’ and ‘day’ ultimately share the same root. Think Latin ‘dies’ (day) and ‘deus’ (god), or if you want modern Spanish ‘dias’ (day) and ‘dios’ (god), or Hindi ‘din’ (day) and ‘dev’ (god).
These all go back to a Proto-Indo-European (PIE) root which does indeed mean ‘to shine’; i.e. the word ‘god’ essentially meant ‘shining one’. In Indo-European culture, the “shining ones” referred specifically to the sun and moon; physical/”heavenly” representations of the (sun) god and (moon) goddess.
It’s quite possible that in Semitic culture the two root words ‘il and ‘ill are related in such a manner – i.e. “shining ones” as referring to the sun god and moon goddess.