Part of the problem is the lack of prestige in replication. What gets researchers promoted is finding novel methods, so most researchers don’t want to spend their precious time on replication.
There is prestige in what gets you recognition, so if journals and departments started recognizing contributions to replication research, like giving them dedicated space in journals, then the prestige will follow.
Fair enough, but the incentives don’t align. Novel research attracts funding, so every if replication gets prestige in the future the business side of research will still run counter to it. A researcher would still gain more prestige work novel work that attracts substantial funding.
Most novel findings don't replicate, so a researcher at Yale can gain plenty of prestige by debunking the sensationalized findings from Harvard, for instance.
This is very context dependent so such a broad generalization probably goes too far. It seems like the “soft” sciences have a much bigger replication crises.
But that point aside, your Yale vs. Harvard example doesn’t fix the funding issue I alluded to.