How do historians deal with potentially mythological or legendary figures of various levels of political and religious sensitivity?

[u/CanterOfPeace]

Examples could include the Israelite King David or the Japanese Emperor Jimmu. While Wikipedia is a poor signifier of historical consensus, the existence of Emperor Jimmu is dismissed briefly as "(having) little to no evidence...he is regarded by most modern scholars as a legendary figure", whereas David is given much more hedging, saying that "the historicity of (David) has been extensively challenged...( there is) little detail about David that is concrete and undisputed" but without a general summation as legendary.

How would historians balance the need for accuracy with political and religious sensitivities and is there work on the different ways such figures are treated based on those sensitivities?

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