My sense has been that hospital administration does not view academic management experience the same as hospital management. At least as far as nursing is concerned. In some ways, even though it doesn’t always seem like it, they are two different worlds. Healthcare is a business, and nursing education is in a small corner of that business. Managing a facility day in and day out and being responsible for it 24/7 is hospital management. Academic management of clinical groups is not dissimilar, but is looked at as a micro sphere of hospital management and definitely not the same. It seems to be an unspoken “rule” that working only in academia is viewed by some administrators as a good retirement gig after earning your stripes in the career ( at least as viewed by the hospital). I’m not saying I agree with these perspectives per se, but it is the reality of the current landscape that I can see. That is why there are in some hospital contexts overqualified nurses from an academic perspective doing certain roles that they might feel they are under qualified for… The academic possibilities and the workforce realities do not always meet for nurses.
All that to say, if you really want a hospital upper management job, I think you would have to follow the recruiters suggestions and go back to square one.
If you don’t want to do that, you should stay in academia where the bulk of your experience seems to be. All the best!