Cold Emailing Boutiques for an Internship?

Hey guys!

I'm a senior graduating this year with no summer internship experience (had my summer/fall study abroad get cancelled so didn't get an internship sophomore year, had another program cancelled so I'm graduating a year early). My university has virtual international internships and I recently got accepted to the program. I thought they would be more industry-specific (IB, PE, corporate finance were my choices), but they're a bit more broad than I was expecting – so they have business development, finance/accounting, financial planning to name a few. I'm looking for some winter internships at different PE firms (remote), so if I get one of those, I'll probably cancel the virtual international internship program.

I was potentially thinking about doing some cold emailing to some MDs at boutiques in Minneapolis (where I go to school) and ask them if there's any work I could do for them, unpaid. Not sure if this is a good idea or if I should just stick with my university's program and do that instead. Any thoughts on this would be greatly appreciated!

Thanks in advance!