Both things can be true at the same time.
Honestly, I still don't buy that they are both true. If you actually have competent people making sound hiring decisions, the conference affiliation becomes a non-issue. Nebraska could have won at a relatively high level in the Big Ten. The problem is we spent two decades sabotaging ourselves from the top down.
Look at the timeline. This didn't start with the Big Ten move. The crap started in the Big 12. Steve Pedersen butchered the Frank Solich situation and made a mess of the coaching search after that. This is proof that bad leadership can sink this program in any conference.
Then look at the actual transition from Big 12 to Big 10. Bo Pelini won 9 or 10 games practically every year, regardless of whether we were playing in the Big 12 or the Big Ten. The conference change didn't stop him from winning. What actually broke the program was poor hiring decisions. Bringing in Shawn Eichorst and allowing him to create a toxic environment in the athletic department directly led to Pelini leaving. That fractured relationship started a total administrative snowball effect, and we've been paying for it since.
At the end of the day, a conference change is just a backdrop. Bad hires were the cause of the downfall in my opinion.