Americans (specifically gained ground with Noah Webster of the Mirriam-Webster Dictionary (originally titled An American Dictionary of the English Language) fame) changed a lot of words to standardize and simplify their spelling that was still phonetically similar. Keep in mind that many Americans at the time in the country spoke many different languages in different enclaves and that this took place soon after the US gained independence from Britain. Notably Webster learned 28 languages to study the entomology in order to facilitate this standardization.

For example, 'k' was dropped from the end of words like musick and publick, which was already adopted by the British public(k) commonly at the time anyway. Another example was dropping the extra 'l' (L) in words like travelling.

Then again, cheque became check and not chec. The British also use 'check' from the word 'eschequier' in the context of Chess, which is also where 'cheque' in paycheque comes from. A check against the king. A cheque against forgery. So why not 'chec'? Because 'check' was also commonly interchanged by everyone the world over anyway for checque and chèque in business before the United States existed. In business between many peoples, why add another word that may be confusing when 'check' is close enough to what Webster and others were trying to accomplish?

Looking at it through the lens of the time and the context of the US populace, it seems logical as many of the changes were readily accepted by the diverse population of the US. It may not now while merely considering it from today's perspective.