Word spelling and pronunciation change over time. Could be the case here, particularly with a word seldom used outside the knife community? May even be a generational issue. Dictionaries are usually always adapting to such changes, words being their bread and butter so to speak. Hard to argue with your Webster reference, huh?
The following might help to support such a thesis (that of change):
https://www.youtube.com/watch?v=tYx9fw-Le98
Might very well explain why I hear choil pronounced by so many without the h being silent. I noted that the word's origins are unknown, probably dating from the 19th century, when many Irish immigrants arrived during that period. Recognizing that, rather deep into browsing, I noted a comment that could offer an explanation, indicating that lenition in Irish English could be the reasoning found to explain this. Hey, "school/shool" and "schedule/shedule" are great examples of United Kingdom English and American English differences that can probably only be attributed to generational change.
EDIT BELOW ADDED, EXCEPT FOR THE PENULTIMATE PARAGRAPH
On the other hand, there are words like chorus, chasm, chord, chord, cholesterol, chemistry, choreography, christian, christmas and on and on. Most come from Old English, Old French or Latin.
I suspect the word "coil" to be, in it's earlier forms, the initial derivative. Webster attributes first use to 1567, though made famous in Shakespeare's "Hamlet" in 1602. If you've seen some royal cutlery or knives from the 16th and 17th centuries, some would have elaborate, delicate metalwork at the area we refer to as the choil. Imagine a choil that is simple and rounded, then add an inch of scrolled metal (part of the actual blade) heading around or within the choil area, or pointing downward, with circular twist/flat plain, eminating from the blade edge at the front of the choil. The patterns of that little piece of metal were often semi-circular. The definition of coil - "to wind into rings or spirals". Could be that for lack of a better word, blacksmiths started calling this area the coil, later changed to choil, perhaps, but retaining the same pronunciation. It certainly wouldn't have been a common usage word. But with diction(ary) evolution, pronunciation changed, unsilencing the h.
You know, it kinda adds up when you think about it. But the last thing I am is an etymologist.
Don't think I'll change, though. It will, however, no longer be maintained on my vast list of personal peeves, LOL.