If you are looking at private schools, visit early (eg year 3 or 4), and go back. If they ask for your contact details, give them. Some schools, not all, do take note of whether parents have visited, to assess whether they are genuinely interested in the school. Do take the time to do a daytime visit if possible, not just the open day. Do assume that the parents are being judged too. Parents who have taken time to consider and can show why X school is their preferred option for their child will be preferred over parents who haven't bothered to visit.
Whilst Oldbean may dislike the prep school badges, they can and do provide a conversation opening at interviews, so if you have them, then you wear them (especially as your fellow prep school peers will be doing so).
Sorry the badge thing was a bit tongue in cheek. Some of them did resemble pearly kings and queens though <runs away>