Learn French: C'est le musée qui aurait acheté l'artéfact. - It is the museum that would have bought the artifact.

This sentence uses the journalistic conditional (le conditionnel journalistique). In French, the conditional is frequently used in news and gossip to report unconfirmed information, rumors, or allegations. By using aurait acheté instead of a acheté, the speaker indicates that the museum's purchase is alleged but not yet verified as a fact.

The translation says it "would have bought" the artifact. Is that the most natural way to translate this concept?

Not usually. While "would have bought" is the literal grammatical translation of the conditional past, English rarely uses the conditional for unverified news. To capture the true meaning in English, you would more naturally say "It is the museum that reportedly bought the artifact" or "...that is said to have bought the artifact."

Why use the structure C'est le musée qui... rather than just saying Le musée aurait acheté...?

The c'est ... qui construction is used for emphasis. It highlights that it was specifically the museum, rather than a private collector or a different institution, that allegedly made the purchase.

Since the English translation uses "that", why do we use qui instead of que?

It is a common trap to translate "that" directly to que. In French, you must use qui when the pronoun acts as the subject of the following verb, which it does here for aurait acheté. You only use que when the pronoun is the direct object, regardless of whether you are talking about a person or a thing.