I wasn't sure UA still made this promise, but here it is:
LAST SEAT AVAILABILITY BENEFIT
MileagePlus members have the benefit of booking awards without capacity controls on flights operated by United and United Express. If there is a seat available, these members have unrestricted access to book an award, even if it's the last seat on the plane.
Mileage Plus Award Travel
Have you tried calling UA and asking to book the flight with miles? Ultimately, not sure how meaningful the promise is, when they can just charge 10 million miles for the flight. Dynamic pricing has changed (ruined) everything.