Not one other time in the bible is רֶמֶשׂ remes used to describe all the living animals.
Every 𝐦𝐨𝐯𝐢𝐧𝐠 thing that is alive shall be food for you; I have given
everything to you, as I gave the green plant. NASB
The word translated as moving thing is רֶמֶשׂ remes.
Here is a few examples were רֶמֶשׂ remes is used :
Genesis 1:24 - And God said, "Let the earth bring forth living
creatures according to their kind, cattle and creeping things (remes)
and the beasts of the earth according to their kind," and it was so.
Genesis 1:25 - God made the animals of the earth according to their
kind, and the livestock according to their kind, and everything that
crawls (remes) on the ground according to its kind; and God saw that
it was good.
Genesis 1:26 - Then God said, “Let Us make mankind in Our image,
according to Our likeness; and let them rule over the fish of the sea
and over the birds of the sky and over the livestock and over all the
earth, and over every crawling (remes) thing that crawls (remes) on
Genesis 6:7 - And the Lord said, "I will blot out man, whom I created,
from upon the face of the earth, from man to cattle to creeping thing
(remes), to the fowl of the heavens, for I regret that I made them."
Genesis 6:20 - Of the fowl after its kind and of the animals after
their kind, of every creeping thing (remes) upon the ground after its
kind; two of each shall come to you to preserve alive.
All throughout the bible remes is only used as a reference to 𝐜𝐫𝐞𝐞𝐩𝐢𝐧𝐠 things, while other words are used to describe other animals.
The same goes for the instances in 1 Kings, Psalms, Ezekiel, Hosea & Habakkuk, remes always means 𝐜𝐫𝐚𝐰𝐥 or 𝐜𝐫𝐞𝐞𝐩. but has in the majority of translations a more general meaning as moves / moving is, which is a faulty translation of Genesis 9:3.
a literal translation of Genesis 9:3 as Young's literal translation;
𝐄𝐯𝐞𝐫𝐲 𝐜𝐫𝐞𝐞𝐩𝐢𝐧𝐠 𝐭𝐡𝐢𝐧𝐠 𝐭𝐡𝐚𝐭 𝐢𝐬 𝐚𝐥𝐢𝐯𝐞, 𝐭𝐨
𝐲𝐨𝐮 𝐢𝐭 𝐢𝐬 𝐟𝐨𝐫 𝐟𝐨𝐨𝐝; as the green herb I have given to
you the whole; YLT
However snakes and other creeping things are unclean to eat. Just as the second part, not all green plants are for food!