I rarely post in this category because I don’t use the WaniKani study system, but I came across this Level 56 word when I was checking out the page for 食 on WaniKani. I was quite shocked when I saw the meaning given because of how these kanji are used in Chinese, and after checking monolingual dictionaries and various examples of usage on the Internet, including in articles, publications from academic sources, and books, I think it’s safe to say that ‘gluttony’ is not the primary meaning of this phrase, and it seems to be a fairly rare way to use this word. ‘Satiation’ (which is one of the default synonyms), should probably take its place. I’ll explain why below:
(The essential points in each section are in bold for those who don’t have the time/don’t wish to read everything. The bold words, read in sequence, should roughly form complete sentences.)
Monolingual definitions point more towards the idea of 'satiation' than 'gluttony'
Here are the definitions from デジタル大辞泉, which most of the monolingual dictionaries available online seem to agree with:
It is possible to interpret あきる as meaning ‘to be sick (of something)’ or ‘to be satisfied’. However, in order for the two definitions to be consistent with each other (which is more likely since that would allow one to evolve from the other, or both to evolve from the ideas in the kanji), the ‘satisfaction’ nuance makes more sense as it is positive and is logically linked to the idea of living comfortably. Moreover, あきる has two kanji forms: 飽きる and 厭きる. The first is indicated by my dictionary (スーパー大辞林) as being appropriate for the ‘satisfaction’ nuance (already pointing us towards the ‘satiation’ interpretation), whereas the second kanji is also one of the kanji for いや, which indicates unwillingness or displeasure and is not the kanji used in 飽食.
Now, it’s possible that 飽きる is the form that’s used for everything because the other kanji is rarer, but nonetheless, with either meaning, the ‘gluttony’ interpretation is a slight stretch, in my opinion, because even eating until one is sick of it does not necessarily imply gluttonous behaviour, as certain people get sick of eating even if they’re just slightly more full than usual. Furthermore, I actually just played into the ‘gluttony’ interpretation here by choosing a favourable translation: あきる involves being ‘sick’ of something in the sense of いやになる i.e. to become unwilling to continue, which does not require the level of antipathy being ‘sick’ suggests. The amount of antipathy required is lost in translation. That aside, I also think that gluttony tends to be accompanied by taking pleasure in eating a lot, which doesn’t really line up with the idea of being sick of it.
The way that 飽食 is used, and the words that it occurs with, suggest that it refers to 'satiation' or an 'abundance of food' rather than 'gluttony'
That may not be sufficiently convincing, however, so it might be a good idea to look at usage. I’ll start with 飽食の時代, which is the example given by 大辞泉 above. If one were to translate it as ‘the era of gluttony’, then articles like this one, entitled 飽食のかげの貧食 (‘the poor eating habits hidden behind 飽食’), would make no sense because they discuss problems other than overeating. Similarly, in books published as recently as 2009, which come up in Google search results, 飽食 is juxtaposed with concepts like 飢餓 (‘starvation’, ‘hunger’) as the food situations in developed countries and developing countries are compared. Once again, ‘satiation’ or ‘abundance of food’ is the more logical interpretation here, because one cannot say that ‘gluttony’ is universal in developed countries, and it makes far more sense to compare food supplies or how much people are able to eat (because there is a common basis for comparison) than to compare eating habits to phenomena resulting from a lack of food (because there are no common criteria).
Other examples of how 飽食 is used point towards the nuance of ‘satiation’: 大辞林 quotes,「土兵と共に―し勇気十倍す」〈浮城物語•竜渓〉. I have no clue what the source is and can only infer that it’s Classical Japanese literature from the verb form す, but I can say that the phrase translates as ‘to 飽食する together with the local/native soldiers and decuple one’s courage’. I’m not sure that eating gluttonously would strengthen soldiers for battle. Eating very well, to satiety, on the other hand, would likely be quite helpful. There’s also a 四字熟語 that uses 飽食: 暖衣飽食 means that one is ‘warmly clothed and well fed’. I do not think that ‘fed like a glutton’ matches 暖衣 (‘warmly clothed’) very well, at the very least based on the literal meanings of the kanji.
飽 as a kanji does not seem to relate to excess, but rather to 'fullness' and 'having enough'
Speaking of the meanings of kanji, 飽 does not seem to carry a strong nuance of ‘excess’. Here are the kanji meanings from スーパー大辞林:
I think that these meanings quite clearly relate to the ideas of fullness and sufficiency, which is to say the concept of ‘enough’, which is the meaning of the word ‘satis’ in Latin, which is one of the root words of ‘satisfaction’, literally ‘the making of enough’, which is related to the word ‘satiation’, and not to the word ‘gluttony’.
Finally, there are other words for ‘gluttony’ that contain kanji that are more consistent with the concept of gluttony itself, so even if 飽食 clearly did mean ‘gluttony’ in some contexts, it’s probably not the most unambiguous word for it. For example, 暴食（ぼうしょく）, 大食い（おおぐい）and 大食（たいしょく）are all more intuitive choices whose kanji clearly relate to gluttony, or at least to eating a lot/eating ravenously. However, I understand that WaniKani is a kanji learning service, not a vocabulary service, so this is probably beyond its purview and objectives.
To sum up, @Mods, could I possibly urge the WaniKani team to reconsider teaching ‘gluttony’ as a meaning (or at least the primary meaning) of 飽食? The only source I’ve found online that clearly places ‘gluttony’ in first position is JMDict, the source of Jisho’s definitions, and I think I’ve found a good number of counterexamples above just by looking through the first three pages of Google results for「飽食とは」. I know that JMDict was compiled by people who were likely far more competent than I am, but this meaning does not sit right with me based on what I know about the kanji in the word as a Chinese speaker and based on how I’ve seen the word used in Japanese, which lines up with the kanji knowledge I’ve just mentioned. I’m sorry for the very long post, but I wanted to flesh out all the arguments and provide sources so my claims and evidence could be verified if need be. (I hope that the inline summary in bold helped somewhat, at the least.) I’m quite prepared to be corrected by way of counterexample or evidence from other sources, but as things stand, I see ‘gluttony’ as being at best a phenomenon ancillary to 飽食, and not its primary meaning.