I came from a T2 NLU and landed atleast 3 internships in T1 firms through cold-emailing. I had decided to practice in a niche area and emailed almost all the Partners of such firms, but I did customise it detailing about an article that they might have written/LinkedIN post or a case they did. Most of them never replied but in the end I did land three internships, which otherwise would have been not possible.

Thus, I would reccomend to take your chances and email them. Caveat - I once emailed two partners in the same team, both forwarded it to the HR and I got to get an earful from one of the partner. I changed my tactic then and decided to only email one partner from one team, and if I got no response within two weeks, I used to follow-up and after one week of following up, if I did not receive any response, then only I emailed the other partners (P.S. - Start emailing with the youngest to oldest partners, since if the youngest doesn't reply, you can always try your luck with the older one but otherwise is rarely possible).

All the best!

Would it matter if they blacklist you, given you don't have an internship with them either ways?

Partners don't really care. Most of them send your mail to spam or block. Some just forward it to HR. Few actually read your mail and CV and contemplate recommending you. A 40 year old professional has a lot more to focus on than to get a 20 y/o desperate kid blacklisted. That being said, calling them up seems too far fetched and can piss them off- don't do that.

As someone who is hunting corporate internships in T1s, I've been constantly given the advice to cold email, follow up, then call. But some others say that cold emailing gets you blacklisted. And, I know others who have called up partners and gotten internships just by being shameless (for the lack of a better word). In your experience, which is the best way? What firms actually blacklist you for this? (P.S. I'm talking about publicly available emails/contacts)