I hesitate to join the fray here, lol...

but I guess just for the sake of interest and tidbits of information if anyone is curious about
one segment of one random British person's experience of this "haitch" business.
This is what I've noticed with "haitch" in the place where I grew up, for what it's worth.

In the 1960's I grew up in Chingford E4, which is postally the "East End " of London (hence the E in their postal code) but only just -- it is an area of geographically the north-east corner of London that borders Essex county. It's a spot that has for centuries been home to an odd mixture of both rather middle class, well-off and educated people, and also influxes of true "East End" folks coming in from more inner London east end such as Spitalfields, Bethnal Green, etc. especially following WWII when many East Enders were bombed out of their homes.
So the speech patterns in my part of London were quite varied. There were very genteel upper middle class people who lived on my street speaking received pronunciation that sounded exactly like the Queen, and there were families that spoke what most people would classify as Cockney, " cor blimey, luv a duck" -- the whole shebang. My schools, also, had this mixture of speech and class. And when anyone had an occasion to pronounce the letter "H" by itself, as a letter, one person would say "aitch" while the next person said "haitch."
In my own case, it's a bit complicated because I was born in the heart of the east end of London, but grew up in slightly more "posh" OUTER east London, but my mother had a Scottish accent and my father had a Liverpool accent. I have a general, non-specific London accent as I guess I learned to speak from the immediate population around me. I have always said "aitch." But my best friend and her family, who were more of the "Cockney" type of folks, said "Haitch" when speaking the letter H.
I remember being confused by this as a child! Was I saying it wrong? Were they? As a child I was even told by someone (a teacher? I don't know) that
"lower classes and uneducated people say "Haitch" but the correct way is "aitch" "I even heard that the "haitch" comes from cockney speakers believing that one should pronounce the "h" sound when saying the
letter, because that's what they are told about not leaving the "h" sound off the front of
words that begin with H, which they do frequently (it's called 'dropping the h' for example,
'ouse instead of house, etc). One might wonder if this is in fact why some people mistakenly say "Haitch" because they fear that to say "aitch" is to commit the sin of "dropping the H." So perhaps it has all become muddled as to the correct way to say it.
All I know is, for me, this was east London bordering Essex, and both were heard in native speakers, but there seemed to be a feeling that "aitch" is correct no matter where you come from, and "haitch" was some kind of mistake people made no matter where they came from.
Eek sorry this is so long but I just wondered if I could add my own bit of geographical life-experience to the conundrum.