If I should go ahead with making a puzzle of this type, the really big question is, what to call it?
A descriptive name would be helpful, but I can't come up with a snappy name conveying the concept of a word puzzle based on 7 reusable letters.
Following a similar line of thought to that which led to Chihuahua, a possible name is Hexagram. I came up with Chihuahua when I saw that there was a possible nine-letter puzzle where only one word could be made - CHIHUAHUA. There are a few puzzles of 7 reusable letters that have only one (common word) solution, but AEGHMRX, with X mandatory, caught my eye, because one meaning of HEXAGRAM, a 6-pointed star, suggests a visual design for the puzzle. The mandatory letter could be in the middle of a hexagram, with the other 6 letters on the points of the star. Possible disadvantages of the name Hexagram - it seems a bit mathematical and nerdish, which might put some people off, while to some people it might bring to mind the I Ching, leading to expectations of prophecy rather than puzzling.
Another possible name would be Chickadee. Like Chihuahua and Chinchilla, it's a cute little creature. It has 7 different letters, so it would be part of the solution of an ACDEHIK puzzle. However, it wouldn't be the only solution word using all 7 of the letters - there would also be the less endearing DICKHEAD.