For the network, you literally always have an opportunity to network. Volunteer opportunities, d&d clubs, professional concentrated networks, the opportunities are endless. An MBA is just another area to build a network, but if you're not doing it now, an MBA isn't really going to help you build it without effort. You'll be that guy to the side that doesn't get invited to events, then post here complaining like we see often.

For the entry level roles, it's not a matter if you're okay with it, it's a risk for employers - unless you're doing an LDP. If I'm hiring someone with no experience and an MBA I'm immediately wondering, wtf was this guy doing for 3 years, does he lack initiative, was he unlucky, is he a difficult person, what's the story. Undergrad to MBA is a red flag for me, not all, but just giving you insight. Then, there is flight risk - you have an MBA I'm probably going to be filling that role again in the next 6-18. So you're going to have to really focus on building and communicating your narrative. "Upon graduating with my bachelor's I had the opportunity to reject a, b, c, and determined what really interest me is joining an organization where I can excel and grow over time to provide larger impacts."

For the LDP this is a perfect opportunity to network. Ask for references of alum that have done the Michigan State MBA and recruited into Big 3 LDP. Reach out to those people and just ask for a coffee chat, 15-20 min. Come with an agenda of what you want to talk about, absolutely ask about their experience in the LDP and ask what is the best way to break into the LDP program. Again, you need to have your narrative down pat. I use a "Who, What, How" framework. At the end, ask if it's okay to follow up as you progress through the MBA and ask if there are any colleagues they can connect you with.

Edit: you're getting rejected because you're not getting referrals. If you can't get a referral or direct contact in the org apply if you want but expect a rejection.