It is a vast simplification.
But it is not as patently false as you want it to be.
You are indeed correct in assuming that the soviet economy was caving in on itself, however the vast amount of resources required to keep pace with us was a HUGE drain on their foreign currency reserves, it created "false" shortages of necessary living products, etc...
THe spending itself would not have killed a functional economy (that quickly...as the jury is still out on what it has done to ours)...but one so hobbled from poor central planning and a desperate lack of liquid currency...well you can see what happened.
However...that level of spending did indeed speed things along quite nicely.
