Switching audio outputs in GNOME Settings does not switch outputs of existing streams

Usually, turning on my bluetooth headphones will change the output to the headphones. As they're not auto-switching (another bug), I need to switch outputs manually.

Therefore, I open GNOME Settings, go to the "Sound" setting, and change the output device from my USB DAC (for external speakers) to the headphones. But when I do this, nothing happens; audio still goes to my speakers.

I found a workaround by installing and using PulseAudio Mixer and redirecting the Firefox audio stream manually to my headphones. But, previously with PulseAudio, I was able to change outputs from GNOME Setting's Sound, and that is what I would expect with PipeWire too.