My approach is based off advice I once heard about essays for college applications. Overwhelmingly it seems like there is a formulaic approach to them that works really well. I assume this isn't specific to college admissions, but really just people in general who read lots of applications (whether it be for a school or job). Part of the advice I read was that you should stick to the formulaic approach unless you are an excellent, confident writer.
I would say my cover letters definitely show enthusiasm and I explicitly spend probably 25% of it discussing my personal interest in the position/company. But writing in a story format that is well formed, relevant to the job, and interesting is well beyond my abilities. I spend enough time as it is on cover letters that generally result in nothing.
Why not try telling a story that shows some sort of personal interest in the organization that you're applying to? Or, try to focus on showing your enthusiasm for the position.
While you're working at changing your style, you should work on some basic mechanical things in your writing. I know that this is HN, so you likely didn't put much time into proofreading, but sentences like this one hurt me in my soul:
"The second is a few sentences saying that my experience matches the desired qualifications in the job ad, and expressing my desire to always learn new technology and improve my existing skills (because it's usually not a literal perfect match)."
Or, consider this sentence:
"I close by saying I look forward to hearing more about I can help."
Again, I understand that you likely didn't put much time into proofreading your comment, but you missed a pretty important word.