So I noticed a shake in the steering wheel on the '04 Lesabre. Only at about 40mph and up. Wheel shakes back and forth like 1/4" each way. No unusual noises, though.

Got home, checked all the lugnuts and hub nuts (all OK). No visible missing wheel weights or tire bubbles/flat spots/wear issues. Jacked up the front. Right front wheel has no play. Left front wheel has a tiny bit of play in all directions. It feels like a shaft in too large a bearing, if that makes sense, and even "clunks" a bit at the end of the play.

Sound like a wheel bearing? What happens if I let this go a few weeks?

Left side sounds like a bearing to me with the clunking.
I wouldn't plan any big trips but around town you should be fine for a while, just watch things don't get worse.

I'm curious, what would happen if the bearing completely failed.

It is likely it will make so much noise, you won't let it completely fail. I have had a front wheel bearing make a lot of noise. When I replaced it, I took the noisy one apart. There was nothing visibly wrong. Bearing tolerances are so critical, it can be difficult to measure any problems with failed bearings (of course I have seen some destroyed bearings that were obvious, but they were not front wheel bearings).