The argument in the OP says "a subject". The Euthyphro criticism applies to all subjectivist views. All of them.
It is you who has decided that because everything you've read on the internet says 'God' that the OP must say God too. It doesn't. The argument is addressed to any and all attempts to identify moral values and prescriptions with those of a subject, an agent of some kind.
Now, my point, for the umpteenth time, is that the argument actually applies to everyone. Which implies it is faulty. It is faulty.
Anyway, why do most contemporary moral philosophers think it refutes the 'God' version of subjectivism? Because if you just stipulate that the subject in question cannot, of necessity, change, then you haven't explained why. And, on the face of it, the stipulation seems false. If God is an agent, why can't he change his mind? No good just saying "oh, well I've defined him as unchanging". Again, get over the childish megalomania and realize that saying things doesn't make them so (well, there are exceptions, but meh).
It isn't consistent with omnipotence. I do things in accordance with my nature. So do you. For whatever I do, it was my nature to do it - my nature being partly constituted by the things I do.
But I'm not omnipotent. Nor are you. So, 'doing things in accordance with your nature' is not sufficient for omnipotence.
An omnipotent being can do anything he god damned wants. Including prescribing today what he proscribed yesterday. I mean, even I can do that - you saying being all powerful involves being able to do 'less' than I can do? Wow, good definition of omnipotence!!
So, you just have to insist - apropos nothing - that God's values and prescriptions are necessary and not contingent - which anyone can do about anything.
It is intuitively obvious that no mind values anything or prescribes anything of necessity. If you think otherwise, explain. Don't just give me a definition, give me a representation of reason that implies what you say. Note, a representation of reason, not a representation by a priest.