Tosafos writes that the gemara takes for granted that everyone agrees that there is an obligation to be מוסיף מחול על הקודש for every shabbos and yom tov. The machlokes between R. Yishmael and R. Akiva is what the source of this is. R. Akiva holds that the pesukim speaking in the context of shemittah require tosefes, and from there we learn to shabbos, yom tov and Yom Kippur. R. Yishmael holds that the source is inuy by yom kippur and from there we learn to everywhere else.
Tosafos asks on the gemara in Pesachim which only considers the bein hashmashos time of Y.K. to be forbidden in melacha, but melacha should be forbidden even before bein hashmashos to fulfill tosefes? Tosafos answers - ושמא משום דסגי בתוספת משהו נקיט, meaing that since tosefes can be fulfilled by abstention from melacha for an insignificant moment, we don't bother mentioning it as a separate entity from bein hashmashos. On the other hand, Tosafos in Beitzah 30a writes that since bein hashmashos is assur m'safeik, clearly the mitzvah of tosefes would be coming to add some significant amount of time onto the bein hashmashos, since it would be impossible to do melacha up to the split moment before bein hashmashos (and it isn't plausible that women violated this as the gemara says). The Ran in Tosafos also concurs with Tosafos that the tosefes must be some significant amount of time, otherwise it would be a safeik issur d'oraysa since bein hashmashos is considered a safeik. Regarding the amount of time that would need to be added for Tosefes, the M.B. and Chayei Adam assume it must be about 15 minutes. The Chayei Adam (Shabbos, 5:1) writes that bein hashmashos together with tosefes will add up to approx. a half an hour. Since he would assume that bein hashmashos is 3/4 of a mil, which is a little under 15 minutes, he considers the time for tosefes to also be approximately 15 minutes. However, Rav Moshe (Dibros Moshe on Beitza) disagrees with this and holds that the time for Tosefes at least regarding eating, even according to Tosafos in Beitza and the Ran who require a significant amount of time for tosefes, would only be a few minutes. Rav Moshe calculates that according to Reish Lakish who holds that eating a chatzi shiur on Y.K. would not be a Torah violation. Therefore, if one ate a koseves ha'gasa in more than a shiur of achilas pras ON yom kippur, they would not be violating a Torah prohibition. Certainly, if they ate that before Y.K. they would not be violating. Therefore the shiur tosefes for which we are demanding one abstain from eating must be within a shiur of achilas pras of bein hashmashos.
Regarding the proof of the Ran and Tosafos that there must be a shiur for tosefes if one holds safeik d'oraysa l'chumra min hatorah, otherwise the time of tosefes would anyway be assur as a safeik d'oraysa. The Magen Avrohom O.C. 342 (cited in maharatz chiyus) writes that perhaps we would not apply safeik d'oraysa l'chumra to bein hashmashos because there is a chezkas heter of it being daytime. Therefore, even if the Tosefes literally went into bein hashmashos, it is necessary to require Tosefes because otherwise it would be mutar m'doraysa even according to those who hold safeik d'oraysa l'chumra is min hatorah.
Going back to the machlokes R. Yishmael and R. Akiva - The Gilyon HaShas asks that according to R. Akiva that the source for tosefes is shevi'is, that would only be a source for issur melacha, but would not be able to create a zman of tosefes from which one must abstain from eating before Y.K.? Furthremore, not all melachas are assur before shemittah, so how can we learn from there that no melacha can be done in the z'man of tosefes shabbos? Another fundamental question that I have is that Rashi says the nature of the tosefes by shevi'is is not that we extend back the issur melacha to a z'man of tosefes, rather we forbid the type of melacha that will help during the shemittah year (Tosafos 9b d.h. u'mutar) also says that the reason we assur plowing is because it is benefiting the land for shevi'is but as counterintuitive as it seems, planting is permitted up until 30 days before R.H.). This is fundamentally different than the tosefes melacha of shabbos and yom tov where the concept is to extend the kedusha of shabbos back to the z'man tosefes thereby forbidding all melacha. The Rashash cites the Rashba who says that R. Akiva would also need to learn from inuy to teach the din of tosefes for Y.K. and melacha on shabbos and Yom Tov and could not just use the pasuk for the drasha of eating on erev Y.K.