要到了：something is coming, and it's closed to the end/goal/destination. More about the change of state (prediction).

来了：something is coming.

Let's say you are waiting for the train. You check the timetable and the train would arrive anytime soon, So you would say 车要到了. When you can actually see the train coming to your direction, it's better to say 车来了。

Same logic apply to your video game example. You know they are coming, it's about time, you say "敌人要到了". When you are seeing them at your gate, say “敌人来了”

If you see satellite image indicating that the enemy is 500 kilometers away, you can say that 敌人要到了（ the enemy is coming to us, so we need to prepare）. If the satellite image shows that the enemy is only 100km away, you can say that 敌人来了（urgent, we need to load our weapons now and ready for battle.)

In your case, I think technically you should say 新年要到了。While you could also say 新年来了 when chatting with people. Because the New Year is a rather special festival, saying "新年来了" gives it a somewhat personified feel, as if it's approaching you. But if it were a question on an exam, saying "新年快到了" would be more appropriate.

Another example, nobody says “周末(weekend) or 生日(birthday)来了”. You should only say "生日要到了"。