It takes a lot of effort to pronounce an h. Just try saying "Her hammer hit his head." I'm sure someday it will be lost in English too. It's already quite common for some speakers to leave the h in difficult clusters like "human", "huge", "historical" and in frequent pronouns "Give it to 'im", "Where is 'e?" The letter h itself has lost its h.
Reducing unstressed
him to
'im is as frequent as reducing unstressed
them to
'em and this is not a sign that
th is disappearing but is rather a peculiarity of unstressed pronouns.
I have never encountered
h-dropping in
historical except for with speakers of a dialect where
h-dropping is a general feature, as, e.g. in Eastern London accents. The
h-less variant of
herb as it is often heard in North America is the original English pronunciation and the
h-restoration in writing is an artificial re-Latinization and the pronunciation variant with
h is spelling pronunciation. There is no general tendency in late Modern English to drop
h but rather to restore etymological
h, as in words like hotel or hospital, which are practically always pronounced with an audible
h today.
Human and
huge start with the consonant cluster /hj-/ which is realized either as [j-] or as [ç-]. Such simplifications of onset clusters are common as, e.g., in
knee, write and
pneumatic. And these simplifications as not a sign of
k, w or
p disappearing.
As for the pronunciation for the name of the letter, I believe in Ireland
haitch (rather than
aitch) is the prevalent pronunciation. I hear the variety
haitch quite frequently in the London area as well but there it is probably a hyper-correction. The
h-less standard variant of the letter name is most likely derived from Old French, where is was pronounced [at͜ʃ] rather than [aʃ] as in modern French. The initial [a] subsequently underwent the diphthongization and the long
a of the GVS, which produced the modern pronunciation
aitch.