For most of my job search last year I was writing cover letters that were basically apology documents. Lots of "I would be so grateful for the opportunity" and "I hope to have the chance to discuss" and generally writing from a position of please pick me. It felt honest at the time because I genuinely needed a job and I was anxious about it.
At some point I got tired and started writing differently. Not arrogant, just more like how I'd write an email to someone I respected but was also an equal. Direct subject line, one paragraph about what I actually do well, one paragraph about why the role specifically seemed like a fit, short closing that didn't beg.
I also stopped applying to things I had only 40% of the qualifications for and started only going after roles where I genuinely thought I could do the job well. This meant fewer applications but it changed how I wrote them because I actually believed what I was saying. My callback rate went from maybe 1 in 25 to closer to 1 in 8 over about six weeks. That's not a scientific comparison, lots of variables, but the shift was noticeable enough that I don't think it was random.
The specific thing that I think helped most: I stopped ending cover letters with anything that sounded like waiting to be chosen. Instead of "I look forward to hearing from you" I started writing something more like "I'd be glad to talk through how my background fits what you're building." Small difference in words, pretty big difference in how it reads I think.
Not a magic trick and obviously the underlying qualifications still matter, but the framing shift felt real and the results backed it up at least for me.