For any English people reading this in England: this advice is both wrong and harmful. If you've done something wrong you're allowed to apologise. You're not admitting liability and it doesn't open you up to further legal action. Not apologising may in fact increase damages against you.

I don't know how other jurisdictions handle this.

https://www.legislation.gov.uk/ukpga/2006/29/section/2

> Apologies, offers of treatment or other redress

> An apology, an offer of treatment or other redress, shall not of itself amount to an admission of negligence or breach of statutory duty.

This is especially true if you're a healthcare professional or work in a healthcare organisation. Your professional registration tells you to apologise; your organisations registration body tells them and you to apologise; your medical defence body tells you to apologise; a bunch of arms-length bodies are clear that you need to apologise if you do something wrong.