That's called decoherence, and it doesn't actually collapse the wave function, it just spreads it out over everything. The measuring device, observer, labaratory, etc all become entangled.

You need an additional assumption that collapse actually occurs to get rid of the world being in a massive superposition we can't observe, if you don't want Many Worlds Interpretation.

Also, physical reality is made up of quantum systems, so there is a real question as to whether the classical is actually quantum. MWI and some other realist interpretations like Bohmian Mechanics would say yes. The Copenhagen Interpretation would say the math is just a predictive tool and we don't know what's actually going on when not making measurements.