Part of the reason for registers carrying powers of 2 bits is the runaway popularity of C and Unix. The PDP-11 had 16-bit registers. Some of the programming conventions we still use today originated with a humble PDP-11 at Bell Labs.

I would put most of the popularity for word lengths that are powers of 2 on the IBM System/360 (1964), rather than the PDP-11 (1970). The enormous popularity of the System/360 is what made computers standardize on bytes, along with words that were multiples of bytes. Prior to that, word sizes were all over the map: 6, 12, 36, as well as really random values such as 23, 27, 29, or 35 bit words.

This isn't to deny the popularity of the PDP-11, of course, but its use of 16-bit words was more of a consequence of the System/360. DEC's use of 12-bit words goes back to the LINC (1962) but by 1970, bytes were obviously the way to go.

What does C have to do with power of 2-bit registers? CHAR_BIT isn’t restricted and could be 7.

Doesn't the standard state CHAR_BIT has to be >=8? If could still be, say, 11 of course.

And to extend that a bit for the OP also read: https://en.wikipedia.org/wiki/Computer_programming_in_the_pu... and https://en.wikipedia.org/wiki/Punched_tape and similar literature for the "interface" with a computer. No "terminal" required :)

And then later versions of the C compiler were written in earlier version of the C compiler...

On top of that, C came about specifically because Ken Thompson and Dennis Ritchie wanted a better language for implementing Unix utilities and, later, Unix itself. https://en.wikipedia.org/wiki/C_(programming_language)#Histo...