No Vibration Over Bluetooth But Rumble Works When I Plug The Cable In

Nothing is broken, and this is worth explaining properly because almost nobody writes it down.

Rumble on a web page goes through the Gamepad API's vibration actuator, and that only works if the operating system exposes a writable output channel for the device. Over USB the DualSense presents a full report that includes the rumble and haptic output, so Windows exposes it, the browser can reach it, and our test fires both motors. Over standard Bluetooth the DualSense presents a reduced report. Windows reads your inputs from it quite happily, which is why every button and stick still tests perfectly, but there is no output channel for rumble. The browser then has nothing to write to and the test does nothing at all. Silence from both motors rather than weak rumble from one is the signature of exactly this, and it is what tells you it is not a motor fault.

Your PS5 works wirelessly because the console does not use standard Bluetooth for this. It uses Sony's own protocol over the same radio, with the full output report intact. Your Bluetooth adapter is not at fault and no driver will change this.

So the practical rule for PC: if you want rumble or haptics from a DualSense in a browser, use the USB-C cable. Test wired, play wired if rumble matters to you. Note that this is specific to the DualSense and DualShock 4. An Xbox Wireless Controller connected through the Xbox Wireless Adapter does expose rumble, and over Bluetooth it varies by Windows build, so the same symptom on an Xbox pad is worth testing both ways before you conclude anything.