Bluetooth pairing is a one-time introduction. Once two devices have exchanged keys, they remember each other and reconnect automatically whenever both are on and in range. Most connection failures happen because that memory is either missing, out of date, or pointing somewhere else.
Accessories are only visible to new devices while they are in pairing mode, usually shown by a flashing light. Outside pairing mode they quietly look for the last device they knew. That is why headphones can seem invisible on your laptop while happily connected to a phone in another room.
Range matters more than people expect. Bluetooth is designed for a few metres and is easily blocked by a body, a bag or a wall. Audio that stutters when you turn away from the source is usually a range and obstruction issue, not a fault.
Battery level is the quiet culprit. Radios need more power than status lights, so an accessory can switch on, show a light, and still lack the power to hold a stable connection.