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"But it may have been Kelly rather than Trump who did the pushing"
I'd say that is likely: in his last interview Bannon thundered that the US was in a "trade war" with China, and also that there was no military solution to N Korea. The former is, hopefully, a load of tosh, and while he may well be right about the second, in fairness even to Trump, it doesn't really do for a strategist close to the president singing off a different hymnsheet on that, particularly as it was said almost dismissively in the context of the N Korea issue distracting the US from its real enemy (China). I can't see Kelly standing for that, and it would certainly not be in the president's interest to have it being mouthed off.