I'm not sure there's a material difference, though.
> You don't become more extroverted, but you can improve your social skills.
What's the difference, though? Having well-developed social skills is an aspect of extraversion.
> You don't become less neurotic, but you can learn to manage stress, or just avoid stressful situations altogether with proper planning.
Isn't this the same thing as being less neurotic? If the overall effect is lower stress when confronted with the same situations and stimuli, then hasn't your personality changed?
> You don't become more agreeable, but you can learn to become more considerate.
Being considerate is just an aspect of agreeableness. If you become more considerate, then you do become more agreeable.
> You don't become more conscientious, but you can make a framework to complete your tasks: schedules, checklists, etc...
I would argue that the act of making those schedules and checklists in order to ensure you get your tasks done is, in and of itself, an act of being conscientious, so teaching yourself to do those things does indeed make you a more conscientious person.
> You don't become more open, but you can educate yourself on opposing viewpoints.
What is the goal of educating yourself on opposing viewpoints if not to understand different people better? You may still disagree with those viewpoints just as strongly, but the simple fact of a newfound willingness to educate yourself is an increase in openness.
To me, you are focusing on the mechanism here, and not on the outcome. After reading this article I found and took a big-five test[0] and found that some questions did try to assess how you feel about aspects of yourself, but many (most?) are about outcomes in your life. That is, "do you feel like it's easy to complete tasks?" is a different question from "do you tend to complete tasks?" The first one is about how it feels: even if you do tend to complete your tasks, you may feel like it's only because you've created your task-completion framework, but it takes concerted, difficult effort to stick with the framework. The second is about the outcome: you do complete your tasks, and the mechanism behind it isn't relevant. I think both things (the feelings and the outcome) are important to assessing the personality traits, to be sure, but changing the outcome (even if the feelings have not changed at all) is a material change. And I'd suspect that, if you do succeed in changing the outcome, the feelings will change over time, at least to some extent.
[0] https://bigfive-test.com/ (doesn't require some bullshit registration to see your results)