This like most things on the internet this is bollocks (or more fitting, horse shit). Obelisk didn’t conceal food, the pigeons fed on crumbs from food given to the horses by tourists. Any pigeon was killed by accident.
This lie comes from this article https://www.thefield.co.uk/country-house/queens-horses-black-beauties-knightsbridge-31908 which also lies about a horse called "Boomer" running away from the escort of Prince Charles and Lady Diana wedding in 1981. The horse was called Wildfire and ridden by Shotgun Mills and while it ran away, they didn't rejoin the escort after having a drink. The escort was also going towards St Paul's which is the opposite direction to Whitehall
Both those horses were specifically Blues and Royals horses