Let me answer your question in two parts.
I am wondering about how much weight the PS has in a PhD application. If it is not that critical, then I tend to write later and spend less time.
This is not a good way to think about a PhD application. A competitive program may have several hundred applications and perhaps ten spots. In such an environment each part of your application should be as strong as you can make it. While some of the folks competing for spots in the program might weight their effort like you suggest, there are enough good people that will produce thoroughly polished applications that one with an obvious deficiency will not make the cut.
Deadlines for most programs begin November 1st and end sometime in mid-December. You will be judged strongly if you cannot write a good personal statement and research proposal within the next three months (as of the time I write this answer). Start writing some drafts now, get plenty of feedback from good writers you trust, and you'll be fine.
Second, what is the personal statement for?
Zeroth, to demonstrate writing ability.
Academia is about continuously justifying and advocating for yourself, in writing. This happens in grant applications, fellowship applications, the papers you write, cover letters, emails to colleagues, Twitter posts, &c. If you cannot write in a compelling fashion, grad school will be harder.
But, primarily, to contextualize your application.
This is especially true if your application has "deficiencies". Your CV might show a multi-year break in research or your research proposal might be somewhat under-developed. Perhaps this is because you were a single parent at some point, or had a medical condition you needed to sort out, or you're first-generation and didn't have good advising during/before/after your undergrad, or you're switching fields, or you decided to take time off to travel the world.
All of the above possibilities, and more, can lead an application to look academically weaker. But, with the appropriate "spin" a personal statement turns your weaknesses into advantages. You learn things through adverse experiences and they can drive your desire to pursue a particular path.
Will I take the straight-A student from a wealthy New Jersey background who happens to have an interest in anaerobic digestion, or a slightly less stellar student who grew up having to dig pit latrines and wants to research ways of improving them? The latter has a lot of personal motivation and context that may lead them to do better research, but this is not easily expressed on a CV and may not be appropriate in a research statement.
Second, to show something of your character.
Grad school can be hard. Graduate students as a population have high rates of depression and work, often alone, in environments with poorly-defined objectives, competing priorities, and minimal oversight, often while facing financial stress. 50% of PhD students leave (power to them!) their programs. Completion of a program does not necessarily lead to personal enrichment or dream jobs.
A personal statement which can speak to one's resiliency and goals provides evidence that a potential admit will be able to handle the challenges described above (whether directly or by finding the support they need and advocating for themselves).
Both of the above reasons are why a program might want, as you say,
the personal statement [to] emphasize the obstacles for one's achievement
Third, to show something of your personality.
Ted put it well in his answer: if someone walks away from reading your personal statement thinking, "I like this person" or some variant thereof, you've done well.
Finally, as an argument.
Your research proposal may or may not have room to discuss why, intellectually, you belong at your chosen program. But your personal statement surely has room. Personal statements tell the Story of You, and that story explains why, logically, your chosen institution is the right place for you given what you've done and what you plan to do in the future.
The personal statement and research proposal complement and support each other in making the argument that what you're doing makes sense, that your chosen institution is the best place for you to do that, and that you being there is of mutual benefit.