Second order RLC filter and load resistance choice
I think your on the right track - poles it is!

In filtering, you generally want some region with as little loss as possible and other region with as much loss as possible.  In your bode plot each low pass pole will create a knee in the curve where the slope drops by 10 dB / decade.  Therefore for a multiple order filter, having all your poles at essentially the same frequency will give you the steepest knee separating your regions of passband and stopband.

Note that your denominator can now would only have a +1 (not +2) if you compared the output of the source (passed the source impedance, Rg) to the final load output.  With an end of +1, your denominator can be factored into (s*sqrt(LC)+1)^2, and voila you have two poles at exactly s=1/sqrt(LC)