The author writes: "Is there a hypothetical world where I would be an academic working on the Birch and Swinnerton-Dyer conjecture right now? Sure. It's probably a world where high-flying students are given, upon graduation, some sort of "mini-Genius Grant". If I had been awarded a five-year $62,500/year grant with the sole condition of "do research", I would almost certainly have persevered in academia and — despite working on the more interesting but longer-term questions — have had enough publications after those five years to obtain a continuing academic position. But that's not how granting agencies work; they give out one or two year awards, with the understanding that those who are successful will apply for more funding later."
These awards certainly exist; the Miller institute at Berkeley, the Simons fellows in NYC, and the Clay Fellows are a few examples. (Some not quite for 5 years, but more than 2. Also, the pay is more like 90k.) The NSF math postdoc is also close to this (2 years full funding if you elect to not teach those years, plus a third year where you have to teach.) Regular NSF grants and NSF CAREER awards are also longer, IIRC, though perhaps less relevant to fresh PhDs.
So, in fact, these "mini-Genius Grants" do exist.