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But which "ch" sound? "Ch" as in "church" is just "tsh". "Ch" as in "charade" is just "sh".

Seconding this. C should be the ʃ sound, and then TC should be the "ch" in "church." The fact that there's no one letter for ʃ is the real tragedy.

I imagine integrals would make a loud static-y burst of noise.

No, crackle is the 5th derivative, not the integral.

It's not an integral sign. It's U+0283 LATIN SMALL LETTER ESH.

Post-alveolar affricates are phonemic in English and deserve their own characters.

(To put it another way, most native speakers treat "ts" as two sounds but not "ch")

Luckily there are other wasted characters, like "x" and "q".