Why does this rising edge detector using a capacitor and a resistor work?
In the video the circuit worked as if the negative pulses were nonexistent.

Yes, that effect is due to the chip. Let's have a look at the SN74LS08's datasheet and look at the circuits on the chip:



Notice the two Schottky diodes between the inputs A and B and GND.

These diodes are needed to protect the sensitive transistors in the chip. Nearly all chips have such ESD protection diodes.

These diodes will conduct when the voltage on the inputs becomes less than about -0.2 V

That "clamps" the voltage to -0.2 V, the voltage will not get much lower than -0.2 V.

In your simulation, add a diode and observe the same effect!