Armenia and Azerbaijan had a war. Probably since the end of the USSR though.

Exactly. Once the occupying force was removed from those countries the underlying ethnic conflict resumed openly. Similar things will happen elsewhere as US military power wanes.

The saying was at least trying to claim something different. The claim was, more or less, that democracy/capitalism (people making this claim didn't distinguish them, mostly) brings peace through freedom and prosperity.

It was more a claim about Western European countries, which, propaganda aside, were not puppet-states of the US.

Cold-war era US puppet-states were different, and at the time mostly didn't tend to have Macdonald's, at least outside of a capitol. We were busy subverting their democracies, and civil unrest doesn't go well with Big Macs.

Yes, and that claim is wrong. Germany has had 50,000 US troops stationed in it for 75 years and the most rebellious thing they've done is politely ask for their gold back (US said no). No need to subvert a country you already have firm control over.