One issue with this idea is that no navy worldwide pursues a calibre larger than 127 mm/5 inch with such a rationale as the primary (!) reason.
Its also worth noting, at least from what I am told, that most services in fact prefer something below 5 inch for AA work, the 76 mm Strales regularly getting emphasised as being very capable because of the sweet spot in RoF and ammo size. The closest Chinese analogue being further refinement of the 100 mm gun family I guess.
End of the day, while I wouldnt dismiss AA capability, I am not sure its a driving motivation. But your guess is as good as mine.
What is the driving motivation then for 155mm?
Land attack with standard unguided 155mm artillery rounds seems like a very poor reason to develop a new, larger gun calibre.
And if you want to go with longer range and/or guided rounds, the cost balloons. So much so that they become much more expensive than glide bombs or low-cost cruise missiles which have much larger warheads.
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The Italian 76mm makes sense, if you assume that the ammunition is a low-cost unguided round, so you want to fire as many as possible. We can see the Italians prefer to use the 76mm over a Gun CIWS.
But in the past 10 years or so, electronics miniaturisation means artillery rounds can now be guided and also see drastically reduced costs compared to previously.
I'm reminded of the Pantsir SAMs ($100K? Each). So perhaps you can take a 155mm round and add Pantsir RF guidance and/or IR-guidance.
If you assume a 2 minute engagement window, that should be an additional 20+ shots against incoming subsonic cruise missiles. Note that the HQ-10 only has 21 ready SAMs for SHORAD, so there is the possibility of running out. A 155mm round should have a lot more range than an HQ-10 and you can have a lot of 155mm rounds, both guided and unguided
So you could have the following air- defence zones:
a) To 2km. 30mm Gun CIWS. Ammo cost is negligible.
b) To 9km. HQ-10 SAMs. (RIM SAM cost $1 million each)
c) To 30km. 155mm guided. (Starting from $100K?)
So you can see that you want to launch a lot of low-cost 155mm guided rounds at long range, rather than use HQ-10 SAMs
Since a destroyer is going to have a main gun anyway, perhaps it is worth going with a larger calibre (which shouldn't take that much extra space/weight) which is more effective in short-medium range air defence. As a bonus, it will also be better at land attack as well.
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Remember that it is only in the past 5-10 years that navies have taken air defence seriously.
Previously the US Navy was overwhelmingly dominant. Plus 10 years ago, I would say China-US relations were pragmatic/productive, so there wasn't the same drive for improved naval SHORAD.