While it's clear that you can't just say I want to go on this day and come back that and get a SAAver award (of course, AAnytime means you can, of course), if you have reasonable flexibility, it's not that hard.
For example, today I got up and thought it might be nice to go to Italy this fall. Fired up ExpertFlyer, did some searching and found 1 U (for my wife) and 1 C (for me with a SWU) RT late October and early November. I had to pay more than the cheapest flight that ITA could find (~$1k vs. $650) but it was available.
I put it on hold, called the EXP desk, confirmed my wife's SAAver J RT and booked mine. So I am spending $1K for 2 RT J to Italy and getting credit for requal for me.
If I was willing to risk the upgrades, I could have bought for as little as $650 RT but it is worth the difference to me to lock in J.
It's not as easy as it used to be but it is far from impossible if you are flexible.