Why are real-time clocks always 32.768khz?

Post a link to the datasheet, then we can see whether the oscillator is suitable for use with a crystal. Alternatively, get a 74HCU04 hex inverter (note the U in the part number), and use one of the inverters in that, unless that's what you are already using.

On a breadboard, use the power and ground buses normally provided, and keep the wires between the chips and those buses short. Connect a decoupling capacitor directly across each chip, not between the buses.