I don't have to get approval to take 20% time, and I work with a number of people on their 20% projects.
I can also confirm that many people don't take their 20% time. Whether it's culture change due to new hiring, lack of imagination, pressure to excel on their primary project, I'm not sure, but it is disappointing. Still, in engineering No permission is needed.
I can't imagine my former TLM ever taking 20% time, but I can thoroughly imagine him encouraging every single one of his direct reports to do so, unless we were on a launch sprint.
I don't want to equate this with working at Google, because I don't have enough information, but here's my anecdote about good managers:
Twice before I've worked with spectacular managers. They treated their employees well, and acted as great shields against the political infighting inside the company. But both times, the managers were forced out, and the employees that depended on them to get interesting work done ended up quitting as well.
If this is indeed, as the TFA posits, a memo from the top being filtered out by a few good managers, in a short amount of time those managers won't matter; they will be forced, or burnt, out. Neither case is good for the company.
If you need humans instead of robots doing a job, that usually (increasingly, as automation advances) means it requires judgement such that pre-written inflexible rules will be inadequate to handle it sufficiently. Which means you need to rely on the judgement of people applying flexible rules for the best results.