license, sincerity, permit
允 centers on granting permission or approval with sincerity: to permit, license, or allow something in a genuine, straightforward manner.
允 is thought to combine 厶 (a stylized person) with 儿 (legs), suggesting a person standing upright and sincere, which extended to permission or approval. The exact historical development is uncertain.
The top 厶 looks like a private seal or stamp, and the bottom 儿 resembles legs. Imagine a person stamping a document with their seal to grant sincere permission.
For イン, picture an innkeeper granting you a license to stay: inn -> イン, and the innkeeper sincerely permits your entry.