I would suggest you write a email to the embassy asking them if a government shutdown should occur as a result of a budget problem, will the embassy still honor its commitment to meet scheduled appointments. Even with a government shutdown as is possible, the U.S. government can not shut everything. Also, these types of things take days or weeks to filter down. The current situation is basically an increase in the government's level of borrowing, so all it would mean is they can't borrow anymore to run operations. There can be short term savings conducted by the President to ease the credit crunch such as delaying payments to employees of the government like senators, congressman, staff and the thousands of other employees or maybe just stop paying some defense contractors for 30 days. That would release multi-millions of dollars.
So, don't worry!
Everything you say is true. However in political terms, the parts of government shut down will be those that generate the most "noise" to get what politicians want. For example, there was absolutely no need to furlough air traffic controllers because they did not get as big an increase in operating funds as they wanted. CURRENT operations could have remained exactly as is. But snarling up air traffic is sure to generate complaints to congress people and move the situation in a direction some other politician wants.
while a budget impasse carries no REAL threat of immediate shut down for budget purposes (you are correct) they WILL immediately shut some things down to hold us hostage for what they want.
Whether that is overseas consulates or not depends on who is calling the shots in the chess game. That would be Obama in the case of Executive Departments such as DOS. He may decide to shut down NVC and not overseas consulates, which means they could process the cases they have but get no new ones.
Who knows? Nice to fantasize that politics have nothing to do with something that is entirely political. You are figuring this like you would your own budget at home "Honey, If I do not get as big a raise as we expected we will still be able to do everything we normally do" "If we do not get the increase in our credit line, we can still pay the electric bill" all true.
Politicians don't work like that. They squeeze people to get what they want. They squeeze where it hurts most so you scream louder and longer.
While K-1 visas are not essential, they are also not likely to generate much noise, so they may be spared