The trouble is the battery. Let me explain. Led acid batteries will show 13,8 or 13.4 when the battery has been sitting. The chemical reaction on the plates still produces voltage because not all of the plate is damaged and will show good voltage because the load of the meter is so small. When the battery is required to be used it must maintain voltage even though it is damaged inside. Your starter will give a try then because of internal damage it can not produce voltage hence a drop in current. This drop will not turn the starter motor and because the voltage is so low it will not even produce enough voltage and current to hold in the solenoid and the buzzing until the battery voltage drops and even the solenoid can not be buzzed.
You have a short or a destroyed pate or plates in your battery. Take a voltage reading at the battery as you try to start until the solenoid buzzes. You will see the voltage drop off to 4 or 5 volts or nothing.
Been at this for 50 years, it is your battery.