Does Redact PDF really remove what's underneath?

Yes. A lot of redaction tools just draw a black box over the text, which is still there to be copied or recovered. Redact PDF flattens every page you mark into a picture with the boxes baked in, so there's no text layer left to extract. The trade-off is that those pages stop being searchable and the file grows a little. Before you send it, open the result and try selecting and searching the redacted areas.