You checked AWARD availability? (If so, how?) Or purchase availability? (You didn't mention checking a particular fare class -- and the awards are only avialable in one -- or at most a very few -- of the MANY fare classes that make up "economy" -- which leads me to believe you did the latter. On AA they're all in T fare class -- the most capacity-controlled award class there is -- but I'm not sure about UA.)

There is little correlation between the two (except that award availability is never more than purchase availability, but at most airlines it's a small fraction of purchase availability).

I'm not familiar with United's economy award availability procedures, but at many airlines they only allocate a small handful of seats per flight (sometimes as few as one or two!) even on flights with hundreds of available-for-purchase economy seats!

Furthermore, no one who participates in this promo has gotten an award available to book yet. So the question is not even how many award seats are available now, the question is: If you don't get your award (or can't pick your award date) until weeks after thousands of other people have gotten their awards (nowhere in the rules does it guarantee specific timeliness or even relative-to-others timeliness of delivering your award, does it?), THEN what will the availability be?