DS's cycling costs:
Family club annual membership about £45,
free BC membership,
nice road bike that should suit for 3 yrs, w/ pedals = £550.
Spare wheels (this is DH's thing, he commutes daily on them for now, 35 miles round trip = £600).
Helmet, Clothing + shoes around £80/yr.
Bits that wear out (like chains, tubes) = £40/yr.
Each event (maybe 12/yr) is about £3 in entry fees & about £50 in travel costs; DH does some of the events too, his fun as well.
We just splurged & got DS a Garmin (£80, should be good for 3 yrs or more, too).
Another £400 for X-bike or so, plus event costs if he does Cyclo-X in winter. Not super cheap, but not touching £3k/yr, either. Maybe £1100-1200 max /yr on avg, over the 3 yr period? Most of the kit will pass down thru the family or can be resold, too.
tbh, a small number of passionate cyclists have fantastic ££££ bikes but aren't themselves very fit riders, it's like using kit to try to make up for their physical weaknesses.
Private schools only meet up for 35 weeks/yr, not remotely enough to get someone to elite level. Have to join clubs outside, too (what manicinsomniac said).