The US citizen kid isn't required to have N-600.
But I highly, highly recommend it.
Without it, many years later, the kid may have to prove they're US citizen. Without you around and lost paperwork over the years, it could be difficult.
In my opinion, this is one of the best gifts you can give to your child. With N-600, USCIS will register them in their system as US citizen forever.
You can apply for kid's US passport today and make a plan to get N-600 done within a year or two. If you save a little each month towards the fee, it won't be too overwhelming.
In the future, the price of getting N-600 will only go up (fees never go down). The cost of proving citizenship 20-30-40 years later involving archive inquiries, lawyer fees will be much higher than fees for N-600 today.
Good luck!