Down Since Birth - Beauxbatons Edition

What if every magical child produced a spike at birth that let prospective schools recognise them? What if Harry Potter's spike was recorded much farther than Hogwarts? What if other Headmasters/Headmistresses kept a better eye on their charges?Here's the Beauxbatons Version...

Long as hell Author's note that you can mostly ignore:Hagrid told the Dursleys that Harry had been "down for Hogwarts since birth." It sounds like a slight exaggeration, right? Just Hagrid being Hagrid. But what if it wasn't?What if the trauma of birth caused a spike in the newborn's magic that was registered by every school within a certain distance? What if Beauxbatons was also within that distance? Or Durmstrang? Ilvermorny? Uagadou?It's less than 700 miles from Bodmin, in the West Country of England, to Marseille, France.. It's also only 1700 miles from Bodmin to Nordkapp, Norway, which is the northernmost point in continental Europe, and barely 2000 miles to the Svalbard Archipelago. It's 3100 miles from Bodmin to Mount Greylock, the home of Ilvermorny. It's 4000 miles from Bodmin to Portal Peaks, Uganda.Durmstrang accepts students from “as far away as Bulgaria” according to the Fandom App's HP pages. (Svalbard to Rezovo: 2500 miles… fun fact, Point Hope, Alaska is 2300 miles from Svalbard…)Ilvermorny accepts students from “all over North America”... Point Hope, Alaska is 3600 miles from Mount Greylock. Alert, Nunavut, Canada is 2800 miles. Tijuana, Mexico is 2500 miles.Uagadou accepts students from all over Africa, but they use Dream Messengers to inform their students and leave a token, so the 4000-mile distance might be less important than the willingness to accept the Dream Messengers.Hogwarts waits until around a student's 11th birthday to send their letters, so Hermione was aware of Hogwarts for a year before she started. McGonagall was confirmed to receive her letter on her 11th birthday, October 4th. What if Hogwarts (and Dumbledore) shot themselves in the foot by using the 11th birthday instead of something more sensible, like, oh, say… a year before attending? Or even earlier? Mahoutokoro takes day students from the age of 7. What if other schools also have day programs or sister schools like a magical primary school? What if they contacted Harry and found out he was being abused, and removed him from the Dursleys long before his Hogwarts letter came?