The biggest problem with Copenhagen is that it doesn't say if there is a physical collapse or that there isn't one. Instead think of it as a rule of thumb; if you apply the quantum/classical cut in the appropriate spot for your experiment, it will give the right answer. But it gives no guidance where that spot is or what physically is happening.
From the many worlds perspective, it's actually quite clear where the Copenhagen "collapse" occurs: it's when something enters a superposition with Earth.
Copenhagen doesn't say that. See for instance, Schrodinger's cat.