Why should it be disturbing?

Being precious is a relationship between two entities, where one of the two must be sentient.
Considering the phrasing "Human life is a precious thing", my assumption is that the reader (or myself) is the one who perceive human life as precious.

To one self the only precious life is your own. This is baked in all of us by default, our genes make us value life.
But what's the value of other people's lives?

We can evaluate other people's lives value with the benefits they bring us. Therefore, the people closer to us will have more values - because the influence our reality. What they do and how they interact with us also influence their value. If you were to save someone for a fire, you would probably give more value to your son than some stranger.

This maps to reality quite well - just think about how shocked are people for accidents happening in their own hometown and how little they care for people dying in a far away country.

It's obvious that people close to use will be worth much more - but let's assume we want to get more granularity on strangers.
How can we evaluate human life which is not related to us?
We can rank them by the value they bring to society, as that value is something you're going to benefit from as well.
Let's say you're about to run over some people with a car and you can't avoid this scenario. You can just pick which ones to kill. Would you prefer running over a man or a fat man? A man or a woman? A man or a kid? A man or an old man?
I won't get into considerations but you can think in term of cost for the society (pensions vs taxes vs health), the ability to procreate, the ability to benefit from more advanced education in the future.

But let's try to take a step further, let's say we want to employ people to solve a task. Let's become a business.
Their value can be seen entirely related to how well they can solve this task.

My view is that we're just really advanced machines without any will or possibility to stray from our predefined path.
Genes are our codebase, the environment we live in is our continuos input, our actions are a continuos output which feed into the environment itself - feeding back in our input.

We're just really complex processes running.
When somebody dies is just like a process who was accomplishing some work that won't continue anymore. What could have that process done?
If we had a state of all the particles (or strings, or whatever your smallest unit is) in the universe, we could theoretically calculate all the possible states, maybe changing the conditions which could have brought our process to die, calculating what it could have achieved.

Creating machines as advanced as ourselves is just a matter of time and technological advancement.
Once we are able to create trained life at will and in a faster timespan (because everybody can do that in 9 months + 30 years of training), the value of life itself would dramatically decrease. Let's call them robots.

Why should I bother hiring free people when I can just create and kill some robots?
And as an extension, why should I be able to kill robots but not people? What even is the difference at this point?

This would essentially bring the value of all the people we don't care about to whatever the cost of creating an equivalent skilled robot is. Which, given robots can create value themselves, is probably going to quickly get to zero and crash the entire economy.
It would be like having slavery with infinite slaves.

Is human life a precious thing? Yes. For now.