I LOVE this! I have never seen it broken down in quite this way and it makes perfect sense.
Thanks -- this is one of the topics I'll probably cover at some point in my blog.
So then, what do you consider are "the basics" on each event? The RO or ROBHS on floor, I guess. Kips and cast handstands on bars? Is the giant a basic? What about on beam? And vault? Is the FHS the basic or the Yurchenko? What list of basics on each event need the most attention?
EVERYTHING: Aerial awareness -- this is something that gets developed continuously, and it's hard to pinpoint any one skill as the most crucial basic foundation, but if I had to pick one, I'd say a front 1/2 (on trampoline or from a springboard or minitramp onto a resi). Obviously, there are other skills that need to be learned leading up to this, so it's not the absolute most basic skill, but with any higher-level skill that including both twisting and flipping, the best way to orient yourself, in my opinion, is to find some way to turn it into a front 1/2.
For example, a yurchenko 1/1 is a backhandspring with a 1/2 turn at the end followed by a front 1/2. A double-double is two consecutive front 1/2s (each preceded by a quick 1/2 turn to prepare and spot the ground). And so on.
FLOOR: Candlestick (in my opinion the first skill that every kid should learn when they first walk into the gym), handstand (the most important basic skill in the entire sport), and roundoff for everybody.
As for front tumbling,t he most crucial basics depend on the gymnast's style and where they want to go. For most female gymnasts, I'd include front handsprings on the list of absolutely crucial basics, but I consider them not to quite be on that level for male gymnasts or for extremely powerful female gymnasts (who will often have an easier time simply working from a punch). How exactly a gymnast should approach front tumbling varies a lot more than back tumbling, in my experience, so between the front handspring, the front tuck, and the front layout, it's hard to name one as being the most important basic skill.
VAULT: Depends on what the gymnast will be doing at the upper-levels. I'd say either a front handspring or a yurchenko entry -- a gymnast who will focus primarily on yurchenkos really has no need for a strong front handspring, and a gymnast who does handsprings or tsuks has no need for a strong yurchenko entry.
Now, this is not to say that coaches need to choose which of the two a gymnast will be doing really early on -- I think the ideal approach is to train both and see what type of vault the gymnast takes to most easily. In practice, this generally means introducing the front handspring first, as gymnasts can't even START work on yurchenko entries until they have developed a very strong ROBHS on floor.
So I'd say the most basic skill on vault is either a front handspring or a yurchenko entry -- both of which can be broken down much further into their components. It's hard to say which among those components is most crucial.
BARS: For boys, giants trump all else as the single most important basic skill (which of course necessitates strong tap swings as the most basic skill leading up to giants).
For girls, it's a little more complex -- I'd say tap swings/giants and kips/casts. I consider the kip and cast to be ultimately part of the same skill, as you never see one without the other once you get past compulsories.
BEAM: First, a disclaimer: beam is not my forte. I spend very little time coaching it.
I'm not sure what I'd consider to be the most crucial basic skills on beam, as there seems to be so much variation in how you can get value in beam skills. And the biggest and most important part of is not physical skills, but mental ones.
If pressed to choose a single skill, I'd probably go with a split handstand (specifically to train the gymnast to keep the hips squared while splitting).
While I'm at it:
POMMEL HORSE: Circles (duh) and false scissors.
PARALLEL BARS: Swings on all four levels (Support, upper-arm, hang, and basket -- though this fourth one should generally be introduced much later than the first three)
RINGS: Swings, handstands, and straight-arm supports