The signals from the mixer are tiny, so you need to amplify them considerably anyway, and you would normally band-pass filter them during this amplification, using ceramic filters (or historically IF transformers). Attempting to use just a wideband amplifer is going to reduce performance considerably, probably to the extent it wouldn't work?.
Once you've got the amplified IF signals you could presumably then use a DSP to demodulate them? - but you're replacing a simple cheap part of the circuit with a much more complicated and expensive part.