That's a good question. I learned them in roughly that order too, singles, doubles, then paradiddles. But I know there are other schools of thought that would go singles, doubles and flams, because those key rudiments appear to cover the major variations in stick motion. It's a subject that's open to debate.
As far as reasoning behind your list of staples, I'd say that alternating singles are the most most natural, like walking with your hands. Doubles are the beginning of multiple rebound strokes. Alternating doubles are the easiest way to create the illusion of a held note on a drum that has some sustain. And even if they're not explicitly noted, I'd include 5's, 7's, 9 's etc. as "doubles". Paradiddles are the first compound rudiment. They are the simplest combination of singles and doubles and have a lot of potential for rhythmic versatility- especially when adding accents or playing on different surfaces. But I'd include double & triple paradiddles, mill strokes, inverts, etc. in that category. Flams deserve a high ranking as well, since they're the beginning of grace note rudiments. They can add another dimension of sound when played at low volume, as well as dramatic power at high volume. They also open up another avenue of technique. In Pipe Band, buzzes are a staple of our playing, as is dead sticking. We basically have 3 sounds we can make: a tap, a dead stick (where the stick lightly rests on the head after striking to mute the strike and then releases) and buzzes. Now if you call a buzz separate (albeit a dense series of) multiple strokes, I guess we only have 2 sounds.
But for most musical situations other than traditional snare; how necessary is it to play involved flam or drag movements, dead sticking, ratamacues, hybrids and the like? Don't get me wrong; it's certainly valuable to expand your vocabulary and it will add so much to your playing and control of the instrument. But these 3 or 4 basics will go a long way.
In any case, actually in all cases, I recommend getting a qualified teacher to work on these.