if you are shorting across the resistor to achieve a "low" on an input of the SN74HC08N, then you are shorting out the Arduino supply,
Rubbish.
You can quite happily connect a logic gate’s input to ground.
The proper way to do it is to remove the wire, that is connected to +5V, when you want an input to be low. When the wire is connected, that is a high. When you remove the +5V wire, the resistor pulls the input low [thus the moniker "pulldown"
Have you ever worked with logic gates? I suggest you review that entire post and have a think about it.