There are two low pass filters involved in this circuit, one being the loop filter (C5 and 3.6K ohm inside the 565 chip) and the other being a smoothing filter to eliminate the 1070 and 1270 Hz ac from the output. I'm not sure which one you are referring to, exactly. However, in both cases, the only connection between the filter and Vcc is through a few capacitors. Since capacitors block DC, it would seem that these caps can connect to +5V, to GND, or to -5V as you like. This assumes that the power supply rails are well bypassed to ground so that from an AC point of view, +5V, -5V and GND are connected together. I'm not sure why they connect to +5. I see this right on the datasheet for the NE565 as well. Sometimes, there is something inside the IC that is sensitive to transient voltages or small ground impedances, and in such cases, connecting the loop filter or low pass filter caps to + supply may be necessary to avoid some quirky problem. I can't tell if that is the case with this chip or not. On the other hand, maybe the caps are connected to +5 just to make the schematic diagram a bit tidier than taking them to ground.
Its always best to connect as recommended in the datasheet because the author of the datasheet knows more about the quirks of the IC than you or I. However, make sure you have a good sized bypass cap on your +5V rail.
To assist you with debug, I recommend that you first test the NE565 without any connections to the comparator. With power applied, You should be able to see the VCO output at pin 4 and you should be able to vary the frequency of this by varying R4. The frequency should be adjustable to about 1000Hz. Now, when you input a tone, say 1070 Hz at pin 2 through C3, you should be able to measure a DC voltage at pin 7, and a similar DC voltage at pin 6. Let me know what DC voltage you get at these pins, assuming that R10 and R7 have been removed to disconnect the comparator.
This DC voltage at pin 7 will vary up and down as you vary the input frequency.