If the pump went out on my aerobic septic system, would that make my kitchen sink clog up? I have snaked every visual pipe and it is still not draining.

No, snake again. What type and size snake are you using. Where are you snaking from?

Only if the tank got filled. Do you know the capacity? MODAD system? Is any other drain clogged?

Your septic system would not clog your kitchen drain unless it clogged the entire house. Did you snake from the kitchen roof vent? Put out enough snake to reach the base and 15 feet more. Let me know how you make out. Good luck, Tom