Usually when an employer thinks I could be a fit, they call and we chat and see if it makes sense to continue talking. The resume and job application is just meant to be a signal of "hey I might be interested & qualified in what you're doing" if they can't be bothered to put in any effort, it's probably a sign they won't put any effort into making an enjoyable workplace either.
So even in situations where tailoring is needed, it's still a completely unnecessary step to solicit details that are typically found on a resume.
Prior to becoming a software engineer, I had to tailor my applications just as you pointed out. BUT I ALSO HAD MULTIPLE RESUMES READY for each situation, since writing a new one from scratch--each and every time--would've been a completely pointless use of my precious job-seeking time.
> You can list the same skills for every job application and eventually find what you want
Not really, actually. Every hiring manager is different, and if you take the time to tailor your application and resume to the hiring manager you do have a higher chance of success.
> and there's about 10 different ways I can spin my experience based on what's being asked in the job listing
Same thing in tech, too.
> I don't WANT to copy & paste my LinkedIn because that's not going to get me the job
Some techies do the same. The reason why it's less common is the critical difference between tech and your space. (At least what I think the critical difference is.) Many people in tech SPAM their resumes, and thus tech hiring managers have to wade through many applications from people who shouldn't even bother applying. The techies who tailor their application end up having to fight the odds that someone just won't look closely enough at the application to call back.
I suspect that the non-profit space has a narrower applicant pool that self-selects better; thus allowing a hiring manager to pay closer attention to each application.