We need to remove fear. But mothers fear for their kids?

If I have no expectation, I can't feel fear.

Yes. You are right. Fear comes from expectation. Fear of death and eagerness of life is the two sides of the same coin. No expectation, no fear. That is the gist of Buddhism. An arhat does not have exception ("What could have be done have be done."), so an arhat does not fear.

Yet, when I fall off a bike, isn't it fear that protects me from dying?

No. It is not the fear protects you. It is your reaction protects you. And your reaction may have come from fear, but it does not have to. Do you breathe because you fear death? Will a person who want to die be able to hold their breath to kill themselves?

And not fearing dead does not mean "when there are some choices of reactions, I choose the one that will kill myself".

I will given an counterexample. Suppose you see a hole in the road, you should stop the bike. But your fear of not stopping the bike in time may drive you to turn a sudden round. Then you get hit by a car (the driver cannot predicate your sudden round and cannot stop their car in time). Here the fear push you into death. On the other side, a person who do not fear death that much may be more clam, and take a more suitable reaction.

If you die from fallen a bike, it is called a transformation from life to death If an arhat "die" from fallen a bike, it is called a transformation from sa-upādisesa-nibbānadhātu to anupādisesa-nibbānadhātu. (Sorry for using Pali, I do not know the English words).

What causes the death? Not falling from a bike, otherwise why arhat does not die from this? So the cause is fear. It is the fear that drives you to death, in the view of Buddhism.

You may not agree this view. You may think an arhat also die from a falling from a bike. But at least consider there is something different between your death and arhat's "death". From a shallow perspective, at least the arhat "dies" peacefully.

Mothers also often worry about their kids.

It is mother's caring protect their kids, not their fear. Suppose a mother is kept in prison, and she worries about their kid everyday. Is this fear helpful to their kid? Is this fear helpful to the mother herself? No.

And suppose a mother ask a Bhikkhu to do her a favor to take care of her kid for a while, and the Bhikkhu accepts. Do you think since a Bhikkhu pursues fearlessness, so they may treat the kid badly unlike normal people?

Do not mistake fear for caring.

Caring may also cause problems in the view of Buddhism, but this is another topic.

Are they weak for doing so?

Yes. They are weak. But it is not their fault. It is their ignorance (avijjā). And a ruthless child murder is also ignorant, thus not stronger.

On the other side, fear is also a weapon, or a mean. The fear that fear drives you to death and pain may motivate you to search for a way to a relief from fear. So fear also drives you to fearlessness.

Same applies to expectation. Expectation, or eagerness drives you to death and pain. The expectation to get rid of death and pain may motivate you to search for a way to a relief from eagerness.

Buddhism is a way Buddha developed to free oneself from fear, eagerness and death (since fear and eagerness causes death). And it has been verified by a lot of arhats in history. Thus it is highly recommended instead of developing your own way.

If you decide to develop your own way, I recommend the biography of Buddha. You can ignore the details like when or where did Buddha bore, but pay a lot of attention to the process and method Buddha use to develop Buddhism. I would recommend Buddha: A Very Short Introduction by Michael Carrithers, a thin book that is very comprehensible. You should be careful since it is very difficult to develop a proper way. And a wrong way may be harmful. Otherwise people would have found out Buddhism before Buddha was born. However, even if you do not get any result, trying to develop your own way may let you understand Buddhism better. You just need to be careful, like Buddha did.

Fear is caused when you cling to something or endear something. It is not fear that will help you, it is caution or carefulness. We wear a seat belt not because of fear, but because of carefulness. We don't touch fire or heated objects not because of our fear, but because of our carefulness. Unlike fear, carefulness is a form of wisdom. It is wisdom that sees potential danger. Fear can arise as a result of one's clinging towards something when one notices the danger. But it is your wisdom that bails you out of trouble, not fear.

A mother worrying about a child is unskillful. A mother being cautious about potential dangers is skillful. It's the same case when falling from a bicycle.

In Buddhism, there are different kinds of fear or 'good fear' & 'bad fear'.

Bad fear is fearing what should not be feared (e.g. afraid of dying when you are old or terminally ill or fearing what other people think about you).

Good fear called 'ottappa' is fearing what should be feared; such as fearing doing bad actions that will bring a bad results (for example, fearing addiction to drugs).

The Dhammapada states:

317. Those who see something to fear where there is nothing to fear and see nothing to fear where there is something to fear — upholding false views, they go to states of woe.