I believe that's the general consensus on what would happen, except it wouldn't just be military satellites if there were munitions to spare. There's only a small chance that all sides would refrain because they didn't want their own attacked and as soon as one side starts there's no reason for the others not to.

Wouldn’t it be a domino effect where debris from the first few military satellites would destroy pretty much everything in orbit?

You are right. This scenario is known as Kessler syndrome.

https://en.wikipedia.org/wiki/Kessler_syndrome

If it comes to the point of shooting down satellites, do space debris even matter? Humanity will blow itself back into the stone age and no-one will be launching anything new into orbit for centuries anyway.

I know not with what weapons World War III will be fought, but World War IV will be fought with sticks and stones".

Albert Einstein

I can imagine hostilities that are severe enough to involve mutual satellite strikes but not so severe as to involve nuclear strikes. A Chinese invasion of Taiwan might escalate like that, for example.

It's probably one part of their strategy. With that said, Russia has far more sinister and worrisome weapons capabilities than their ASAT program - in the event of an actual war.