Why are local field potentials generally band-pass filtered?
There is no physical limitation whatsoever on the frequency band you wish to record, other than the hardware limitations on the sampling rate.

Often electrophysiological recordings suffer from noise (mainly in the high-frequencies) and drifts in the baseline (low-frequency range). 

In the end, you wish to filter out as much noise as possible, without throwing away the baby with the bathwater. In effect, prior knowledge about the frequency range of the responses of interest are vital. Anything outside that range can, and likely should be filtered out.

For example, FFT analysis of EEGs mainly focuses on bands limited to the frequency range of 1 to 50 Hz. The mains frequency is 50 or 60 Hz, often imposing sinusoidal noise when the room is not shielded. Hence, what reason would there be not to use a low-pass filter with a cutoff around 50 Hz to get rid of it?

Lastly, offline filtering methods are aplenty. My personal preference nowadays is, is to collect the raw data and filter offline. That way you have the baby and the water, and you can focus, with trial and error, on where the baby is and where the water.