"The US would push the USSR much harder in proxy wars if it thought they could lead up to an economic/military collapse of the USSR or its satellites, and all the European countries would participate more in these proxy wars."

Agreed, but there's still peace in Europe in this scenario.

"Yes, but the combined resources of the biggest (say) 1000 companies are far far greater than those of governments, simply because there are so many more corporations."

"Corporations" do not act coherently like a national government does. There is no "United Corporate Alliance" or any such thing.

"This is true both inside a country and summed across Europe. And most corporations by far would lobby very strongly against war inside Europe."

Some would, but some would probably push for it (military contracting can be enormously profitable). There were certainly plenty of companies that pushed for a US war in Iraq.

From Schindler's List:

SCHINDLER: There's no way I could have known this before, but there was always something missing. In every business I tried, I see now it wasn't me that was failing, it was this thing, this missing thing. Even if I'd known what it was, there's nothing I could have done about it, because you can't create this sort of thing. And it makes all the difference in the world between success and failure. [He waits for her to guess what the thing is. His looks says, It's so simple, how can you not know?] EMILIE: Luck? SCHINDLER: War.