Why would you put your input amplifier in front of your filtering for an ECG signal?
Am I missing some important reason why you would do the signal
  conditioning in this order?


Yes you are...

The front-end differential amplifier will be chosen such that it has a common-mode rejection level of many tens of dB, quite possibly in the region of 80 dB.

This diff amp converts a differential signal into a single-ended signal and any common-mode interference will be largely ignored.

If you were to put filters on both legs of the diff amp, to avoid a balance mismatch you would have to choose components (such as capacitors and resistors) that were matched to at least an equivalent level of -80 dB. 

You could regard 1% capacitors as having potentially a difference in value of 2% and that, in dB terms could be regarded as -20 log(50) = -34 dB. In other words, you would never get decent differential common-mode performance with filters on each leg before the diff-amplifier.