Immersion is the cornerstone of AJATT, with the SRS serving as an optional supplement. While Anki or other SRS applications can be useful for helping learn vocabulary faster, it is possible to become fluent without using them. If using an SRS reduces your motivation or enjoyment, it may be in your best interest to forgo using any SRS and instead focus on the core of the method.
To improve learning speed and efficiency, it is recommended to incorporate some form of repetition, provided you can tolerate it. We eventually forget everything that we don't review. While total, mass immersion can partially counteract the forgetting curve, it can't help when trying to memorize more uncommon words.
In the case of Japanese, the SRS proves especially useful because it helps you learn how to read words written in kanji and ensures that you don't forget kanji readings. It is particularly helpul when dealing with rare kanji.
Personally, I have observed that the majority of my failed Anki reviews stem from forgotten kanji readings. I may recall the meaning of a word, but struggle to recall its pronunciation. Therefore, if your target language does not utilize kanji or hanzi characters, it is totally fine to omit the use of Anki.