Would olive-oil bathing (as ancient Greco-Roman cultures may have done) have actually succeeded in cleaning a person?

Oil has the funny attribute that while it feels like it's making you more messy, its slippery properties can help dirt slip away too. So e.g. pouring oil into a clogged toilet can help unclog it. (Sometimes. Don't sue me if it doesn't work.) Or how the oil bath around a combustion engine also helps carry the dust and crud away in addition to lubrication.

So I think it just might, although the process is probably more involved than just pouring oil over oneself.

Olive oil is also a part of some soaps, often a significant part - like over 50%.

Wasn't it more of an oil and scrape process?

Try it and report back