I think your right, but there are two problems. The incoming signal, let's say centered at 100MHz will not exactly match the oscillator. So the resulting frequency won't be 0Hz. The bandwidth of the incoming signal will result in frequencies maybe to 1MHz. So you could lose something, maybe not.
The problem is that if you set the oscillator at let's say 95MHz, a 95MHz transmission from another station will appear at the lower end of the spectrum, only slighty diminished assuming the correct length of antenna. High pass filtering stages would reduce this unwanted station.