And yet the
Count of St Germain seems to have done precisely this, repeatedly, all across the courts of Europe in a way that suggests that travelling nobility would get a presumption of good faith. Hangers on as guests at stately homes are a fairly common character in Austen and similar period literature. Hospitality is an important theme in Arthurian legend. We've even got
modern examples that suggest robust background checking isn't a big thing of people who claim certain backgrounds even today.
I absolutely get your point that it seems illogical, but a player who is more thinking about the kind of conventions that some literature suggests may find a flat denial capricious. An introduction or some proof may be needed, but the Noble background starts with a signet ring and a scroll of pedigree, so they should have that available to them.