Is your wife a naturalized citizen?
If so, yes your kids can serve as President.
If so, yes your kids can serve as President.
It takes years to qualify for citizenship. Does that mean my kids are in some kind of limbo between second-class citizen and full citizen until my wife receives citizenship? What if something terrible happens and my wife dies before she earns her citizenship? My kids are forever designated to a second-class purgatory because their mother didn't earn her citizenship in time?
Do you realize how ridiculous this sounds? Are you aware of how many millions of Americans are born to the same circumstances as my children are? Do you really think that any legal authority in the land will designate these millions of native-born people ineligible to be President --thus turning them into de facto second-class citizens?