You have full manual controls with RAW capability. Regardless of how auto works, tweaking manual settings will produce far better shots than the Nexus 6 would ever be capable of producing.
Now for the helpful less "fan boy sounding" response: LG tend to blow out highlights in the background to produce a proper exposure for the main subject.
Even the G5 looks like this will be the case,
as sampled here
Personally, I think Samsung nailed the auto camera experience with the S6 generation phones. LG have more capability, but it's lessened by the fact most users are not photographers.
Kind of looking forward to the S7 for this reason, it brings back all the features I couldn't tolerate losing with the S6 and enhances the camera even more.
To fix your problem (particularly in environments like your photos)
Turn the brightness up to about 80% with auto brightness disabled
Switch to manual in the camera
Keep everything auto, but fiddle with ISO and exposure time.
Keep the ISO as low as possible while making sure you can take a photo without motion blur from hand shakes (1/30 is as slow as I go usually)
When it looks right on screen, take the shot.
If in doubt and you're feeling technogically empowered, enable RAW.
Happy snapping