Most leaders I have worked with are too far in the past or
will not be interested in me leaving my current organization.
Doesn't matter. I would ask them anyway. How important your recommendations are is context dependent so you want to figure that out first and then take proportional effort to get some.One option is to contact some research group and express your interest in volunteering some of your time. Be open about your desire to gain some experience and get a feel of what it is like to do research, and ultimately get a recommendation letter. My sister did that and it worked really well for her.
P.S. One of the worst parts of application process.