If a full scale war breaks out,if Greek armed forces are in a bad position and request French military assistance,we would have no other choice than to step in,even if with the prospect of French casualties. The entire French political class as well as the public opinion being so anti-Turkey would support any military interventionism to help Greece. Other European countries would also have no other choice but to military assist Greece.
Actually,Erdogan has made so much enemies in the region that a war would be the right occasion for any country to "settle" their feud with Turkey. Greece does have many allies,Turkey doesn't.
I don't expect any full scale war anytime soon at least,skirmishes are very likely though before both sides are pressured to "restraint".
I agree with some of the points you made. But realistically, European countries are not going to line up to support Greece.
Firstly, that'll just give Turkey the green light to fully open up the borders for the refugees. Even with foreign troops deployed it'll be difficult to hold this back if Turkey's Police/Jandarma force-opens the gates. Admittedly, last time they half-assed it. This time they won't.
Europe can deploy all the naval assets it wants. But is it going to go toe-to-toe with Turkey in the designated Mavi Vatan area? I doubt it. Turkey could not only provide AA cover for almost the entire area, but also line up anti-ship and ballistic missiles across the entire coast.
To make matters much worse, Turkey has one of the largest drone fleets in the world. They can monitor the seas non-stop, and soon they'll be armed with anti-ship missiles. Yes, loitering ship-hunters. This can free up Turkey's F-16s strictly for air-to-air combat.
I understand that France's capabilities are among the best in the world. However so are Turkey's. Both countries have their advantages.
But France cannot take on Turkey in the Eastern Mediterranean and the rest of Europe can only offer token assistance.
Turkey has proven time after time that it is willing and able to win battles no matter the cost. I doubt France really wants to lose troops, jets or even ships for little gain.
I'd say that France really just wants to sell some equipment to Greece and maybe take some of those natural resources. Also, prove to the rest of Europe that they are in control. That's about it.