They've historically done variants of both over the years. I think the main point now is that there aren't many empty flights anymore.

A prime example where you do see good last-minute opportunities for award users: airlines frequently open up F award availability in the last days before a flight. Sometimes we grumble about how that seems to be the only way to get the best aircraft types or most desirable routes, but it is a variant on the concept in the OP. When the airline thinks it can sell out F, you need to use a rulebuster award to get the seat. Then 5 days before departure, the airline realizes it has a dozen empty F seats, so the "regular" award and upgrade buckets sometimes show up with a couple seats available.

I personally haven't been able to take advantage of this - I tend to plan my award trips farther in advance, picking through the limited premium cabin award seats available 6 months in advance. But a last-minute traveler sitting on a pile of miles could certainly jump on these.