Standalone ATmega328P stops responding when RX is connected
There are two things wrong with the circuit in that demo.

There is no decoupling capacitor on the power pins. This is required to keep the power stable during opertaion.
More critically: this is being powered from 3.3V, but is being run at 16MHz. That is outside specifications. You should use an 8MHz crystal to operate at 3.3V.