MBB internship pools are very competitive, but there is no single perfect profile. Recruiters are looking for a combination of academic ability, achievement, leadership, problem solving, and evidence that you have made an impact.
A balanced profile can absolutely work if the individual pieces are strong enough and tell a coherent story. What hurts candidates is often not the absence of one elite brand, but a resume that describes responsibilities instead of achievements or gives the recruiter no clear reason to remember them.
Benchmark yourself honestly, improve the areas you still control, and recruit beyond one firm or one tier. You can be a credible candidate without looking identical to the most polished profile you see online.