Beis Shammai says: "We do not carry a child, a lulav, or a sefer Torah, to the public domain (on Yom Tov) ; but Beis Hillel allows this.
The Gemara explains that Beis Hillel holds of the idea: "mitoch" which means that "since it (carrying) was permitted for food needs, it was also permitted for any non-food need as well."
Beis Shammai would reject the "mitoch" principle (at least as far as Beis Hillel applies it here).
Rashi in his comments on Beitzah 3a and Eiruvin 39b, (and this is how the Rosh and Ramban understand Rashi as well) shows that he would hold the following rule:
Any melachah that could have been performed before Yom Tov to the same quality effect, may not be done on Yom Tov itself. This is an issur d'Oraisa (a Torah prohibition and not merely Rabbinic.)
Also, the continuing subject in Beitzah 12a eventually explains that we only say Beis Hillel's application of "Mitoch" in 3 additional cases besides the melachah of baking/cooking:
1) Shechitah - slaughtering an animal
2) Havarah - lighting a fire (from a pre-existing flame)
3) Hotza'ah - carrying
Otherwise, we do not apply the rule of "mitoch".
The simple reasoning is :
Cooking is the one melachah that (almost) never requires one to have done it before Yom tov. This is because freshly cooked food is always considered tastier than yesterday's reheated fare. So, the Torah would not ask a person to avoid cooking on Yom Tov and make the food before the holiday, because such a thing would not taste as good. (True on shabbos we must prepare the food beforehand, but that's why Yom Tov is special. :) )
Slaughter too is always allowed on Yom Tov (certain Rabbinical restrictions apply) and for the same reason. Fresh meat is better. Freshly slaughtered and cooked right away is just awesome compared to defrosted burgers reheated from two days ago!
Lighting your cooking fire is obviously done on Yom Tov because its the thing that makes the fresh cooking happen! If you did it all beforehand (besides your pilot light), there would be times you will need to raise the flame anyway once you get down to cooking.
Carrying for food also happens on Yom Tov and could not be done beforehand. If you need to make fresh cakes to bring to your friend's house for the party, you can only bake them right before the meal, and so you must carry on Yom Tov.
All of the other 39 melachos usually could be done before Yom Tov with the same quality effect. (Sure you can dream up some what if's, but the overall idea of any other melachah is that it can technically be done beforehand.)
(BTW, even if you "could have done it before" there are ways of still doing it sometimes and in many cases on Yom Tov , but that is only through special (allowance) heter and is not applicable to all works, and is not the subject of the OP.)
An example of what could be done before Yom tov would be Trapping a deer. The meat is just as good if you trapped the deer last week or on Yom Tov. So, Trapping on Yom Tov is forbidden, even if you are trapping it for the need of food on yom Tov! However, the slaughter must be done on Yom Tov for fresh meat.
So, in conclusion:
35 of the 39 melachas may be done ONLY for direct food purposes on Yom Tov (if they couldn't be done beforehand). They therefore do not qualify for "mitoch" of Beis Hillel and absolutely may not be done for non-food needs of Yom Tov.
example: Grinding may be done on Yom Tov if the food will lose some of its quality by doing it before Yom Tov (otherwise you should have done it before.) --- grating apples --- chopping onions into fine little pieces.
So, one may not apply "Mitoch" to grinding. Ex.: crushing metal into shavings for an arts and crafts project would be forbidden on Yom Tov because it has nothing to do with food preperation (or enjoyment of the body) and therefore cannot be allowed.
4 of the 39 melachas may be done on Yom Tov in just about all cases as long as it is for a Yom Tov need of some kind, (is for a Jewish person, and is something most people need) even if not for food preparation. The reason is Beis Hillel's "Mitoch". Once they were permitted for food needs on Yom Tov, they were permitted for any Yom Tov need. Example: carrying a book.
The reasoning behind this, is that since these 4 were so needed for food prep. on Yom Tov, Hashem granted them complete heter. That is how to explain the statement of Beis Hillel. Since they were permitted for direct and constant food needs, they were permitted for non-food needs of Yom Tov as well.
The Beis Yosef however, says that Mitoch allows you to do these melachas for no reason at all. (not just for a bona fide Yom Tov need). (see details S.A. O.C. 518)
The way to explain this is that since these four works are so key to enjoying food directly, Hashem simply cancelled them outright, as opposed to the other 35.
One more point: Since the three melachas of baking/cooking, slaughter, and lighting, have very little use other than for food or physical benefit, the only melachah we tend to hear about under the "mitoch" principle is only carrying (even though the other 3 are granted "mitoch"). That's why it seems carrying is always the one given the leap of food to non-food activities. Carrying is the one melachah exclusively that a "mitoch" grant will allow a colorful and large amount of non-food needs to be done on Yom Tov.
As far as the OP's assumption that a stricter view of lighting candles couldn't work; ... this is not an issue. The questions of melachah here are Biblical. Lighting candles for Shabbos or Yom Tov (especially on second day Yom Tov) is a Rabbinical obligation, so it would move aside. You could even have your lights burning with more fuel from before Shabbos or Yom Tov for all 2-3 days in advance. But we don't have to because of "Mitoch" -- Beis Hillel. Lighting from a pre existing flame is always permitted on the 6 Yom Tovs.