Do smartphone cameras generally expose to the right?
That's not something you can generalize because there are lots of smartphones out there with lots of different sensors in them that behave differently. Yours in particular appears to do that, probably because the manufacturer determined that doing so gets the best performance out of the sensor.
In either case, why isn't Lightroom able to figure out the correct exposure when I click the Auto tone?
Correction of a raw image is almost never a simple matter of adjusting the exposure by some constant k and having everything come out right. Camera sensors are nonlinear beasts, and the manufacturer of your handset will have done extensive testing to figure out what kind of transformation to apply to the raw sensor data to make it accurately reflect the light that actually fell on it. The result of this testing is called a base curve.
Lightroom and applications that process raw images ship with base curves for the cameras they support, identify which one to apply based on the make and model information in the raw file and apply it early in the pipeline. Your Nexus 5x is probably not one it knows about, and Auto Tone's reaction to that might be to apply a one-size-fits-all base curve or none at all.
Manufacturers don't usually hand their base curves to anyone who asks. It's possible to use a raw and JPEG taken of a specially-constructed scene to approximate the curve used by the camera when it creates a JPEG. (Here's an explanation of how the developers of Darktable create their base curves.) Adobe is either going through a similar process or may pay camera manufacturers for the information.
The way you'd fix this for yourself is to go through a similar process to determine the base curve and then develop a Tone Curve (Adobe's terminology) preset that mimics the results.