The important thing to realise is that pinyin descriptions of characters representing ancient words are prescriptive in accordance to ancient literature. This means that as long as you have the ancient text to back up your pronunciation, it makes no sense to question your choice.
The first step is to determine what 汎 actually means in this context. Remember that characters are not the same as (spoken) words; one word can potentially be represented by a multitude of characters. In this specific Analects quote, ctext gives a vernacular translation of 汎 as 廣 (broad), and an English translation by James Legge as overflowing.
The second step is to find written records of Fanqie decompositions of this character, which will tell you how to pronounce it. The site that you linked in your question, zdic, normally has a Kangxi dictionary section for any character that you look up. For 汎, we can see that several Fanqie decompositions, along with the meaning of the character for that particular decomposition, have been given:
So, the total options for pronunciation of 汎, as given by the Kangxi Dictionary, are:
Pronounce it like the character 氾, which could mean
- The appearance of floating, extended to mean going with the flow;
- Overflowing (this matches James Legge's translation);
- Gentle, swift, nimble.
Pronounce it like 馮, which could mean
Pronounce it like 乏, which could mean
- Small, quiet sounds;
- The sound of rushing waves;
- Same as 渢 (float). (The rest of the line says that the meaning flood; overflowing is contemporaneously written as 泛. This usually means that by the time of this reference, the meaning flood, overflowing is no longer associated with the character, and the pronunciation bound to this meaning is also dropped.)
From these given examples, the best choice is to pronounce it like 氾, which itself has multiple pronunciations; but a good dictionary will tell you that 氾 is used as a variant of 泛, which is unambiguously pronounced as fàn in the sense of to flood; to overflow.
Most importantly, this only shows that pronouncing it like fàn is not wrong, and does not mean that there aren't other choices available. As long as you can back it up with examples in literature, it cannot be deemed wrong. Unfortunately I could not find an example that shows that it should be pronounced fà, but if an example exists, then it is also not wrong.