Learn Italian: Essendo il disagio evidente, chiediamo scusa. - The inconvenience being obvious, we ask for forgiveness.

You're right that a gerund usually has to share the subject of the main clause (which is noi here, hidden in chiediamo). However, this sentence uses an absolute gerund (gerundio assoluto). This structure allows the gerund to have its own distinct subject, as long as you state that subject explicitly in the phrase, which is exactly what happens here with il disagio.

Why does the sentence start with the verb Essendo rather than the subject il disagio?

In Italian absolute gerund constructions, the word order is usually inverted. The gerund almost always comes before its subject. While in English we say "The inconvenience being obvious," translating that word-for-word as Il disagio essendo evidente sounds extremely unnatural in Italian.

Could we say ci scusiamo instead of chiediamo scusa?

Yes, ci scusiamo (from the reflexive verb scusarsi) means essentially the same thing and would work perfectly here. Chiedere scusa is a very common fixed expression that literally translates to "to ask forgiveness," and Italians use it interchangeably with scusarsi to apologize.

Does the word disagio always mean "inconvenience"?

Not always. While it frequently means a practical "inconvenience" or "hardship" (often used by companies apologizing for service outages or delays), it is also the standard word for physical or psychological "discomfort," "awkwardness," or "unease."