Should I follow up after getting a generic “gone another route” email?

I applied for a position in a company I was really wanting to work for. My partner also works for the same company in a different area and told the hiring manager I was planning on applying. She gave him her business card as well as told him to give her my name so she can keep an eye out for my application. When I applied I sent an email to the one on the card with a brief hello I would love to work for you here’s my info blah blah and attached my resume.

I hadn’t heard anything for almost a week until this morning when I got a “we’ve gone another route with people whose background and experience more closely align” email. (I am literally doing the exact same job at a different company and have been in similar positions for about 5 years now so I’m definitely qualified/ aligned) When my partner had talked with the manager she had mentioned she’d look for me because AI has been auto denying people from time to time… my only thing is wouldn’t that happen right away? I’m nervous about what to email her back with but I wanna know why I was denied and what I could do to be in a position at their company.

This is basically a step in the door to my dream job (like if I wanted to work in a grocery store bakery and got a cashier position so I could chat with all the bakery staff to see what I could do to eventually move over there)

If it’s a bad idea to follow up and ask why I was denied I’ll easily leave it alone but I’m afraid that I’ll lose out on an amazing opportunity because I was too afraid to speak out - any advice?