I did this and got partner referrals. But i was an experienced hire coming from another industry. And the people who replied tended to have similar background to me and felt sympathetic.
I think you're putting the cart before the horse and not helping your chances reaching out this early. Especially if you were already recently dinged or not invited to interviews by same firms.
Caveat: if you're in an undergrad club or biz frat, and have friends or an alumni database specific to that group, they'll be more likely to help you out of the blue.
I'd wait until the summer to reach out at earliest. Will look very bad (imo) if you do it before you've shown ability to do the job well at another firm first. They hate overeager, underqualified undergrads spamming them. They can't help you right now, so why email them?
If you are committed to reaching out now, YOLO. Yes, you'lll get replies. Def not as many as when you're potentially a more legit candidate (in their eyes) in future. But you'll get replies. I'd recommend starting with lower tier and working your way up.
If/when you do email:
1. Make email as short as possible.
2. It's not short enough. Cut 1-2 more sentences.
3. Be honest, explain situation in first 2 lines.
4. In another 1-2 lines: the ask and the why (why you're reaching out to them)
5. Say thank you.
6. A/B test attaching resume and not attaching resume
You have an unfair advantage with access to penn alumni database, especially Wharton. Don't waste your trump card by focusing on prestige so early in your career.
Get work experience, prove you're worth a reply.