Why won't the bandstop filter work in this circuit?
Your band stop filter is made from two other filters - one high pass filter and one low pass filter.
Each of those two filters are of the lowest order possible. They are simple resistor/capacitor filters with a roll-off of 6 dB per octave or 20 dB per decade.
That means that in the stop band for each filter, the output is attenuated (reduced) by 1/2 for each doubling of frequency.
Take the low pass filter (R3 and C5). The cut-off is at 50 Hz.  With 6 dB per octave, it will reduce a 100 Hz signal to half its original intensity.
The high pass filter (C4 and R1) has a cut-off of 60 Hz. It will reduce a signal at 30 Hz to half its original intensity.
The gap between the high pass and the low pass is only 10 Hz. That's only 1/5 or 1/6 of an octave. At best, you are getting 1/5 of 6dB attenuation from your &quot;band stop&quot; filter. That's not enough to see in your frequency plot.
You need a better filter to remove power line hum.
Analog filters to effectively remove powerline hum must use higher order filters. Those can be quite complicated to design and build properly.
I had a look around the internet but couldn't find an example that I thought was really good.

A better way to deal with hum is to filter it digitally.
Many analog to digital converters (ADC) intended for the kind of work you are doing include a digital powerline hum filter that is far more effective than what you have tried to build.
If that isn't effective enough, you can apply a powerline hum filter to your data in software. Once the data is on your computer in digital form, you can do a lot of things to clean it up.
This paper gives an example of implementing an effective powerline hum removal using MatLab. GNU Octave gives you most of the functionality of MatLab for free, and many MatLab programs will &quot;just work&quot; when used with Octave.