“Even if there is no chance of taking Armenia directly, this might be a wedge issue to pry away Azerbaijan (and it’s oil and gas) from Russia.”
Putin could, if pushed, do the same thing to Azerbaijan. Both it and Armenia were once part of the Soviet Union, and before that both were part of the Russian Empire. Putin sees both as within the Russian sphere.
NATO should see that Turkey is in GREAT error, and put Turkey’s NATO membership on the table unless Erdogan stops his Azerbaijan adventure. NATO surely has no interest or business in promoting the conflict and if it led to confrontations between Turkey and Russia directly, NATO nations could be treaty obligated to aid Turkey - which in my view they should not do. The better course would be before things get to that point boot Turkey of NATO.
Should have done that in 1974 when they invaded Northern Cyprus.
It is a tough position for many NATO countries.
Turkey itself has big strategic value - location, a top 20 economy, a top ten arms industry, etc.
Turkey is not inherently the problem - Erdogan and his political party are.
NATO hesitates to throw the baby out with the bathwater, hoping to wait out Erdogan, and somehow manage him in the meantime.
The NATO Treaty does not obligate members to support another member committing aggression - only if they are attacked. It would be a high stakes game for Turkey to try to draw NATO into a Turkish conflict - that might precipitate their expulsion, right while they are in conflict.