Note that Ctrl-A is not likely what bash's "bind" feature uses to move the cursor to the start of the line. Rather, bash responds to ASCII code one.
With many terminal programs, pressing Ctrl-A will result in sending ASCII code one to the foreground program. However, it seems your terminal program is using Ctrl-A to accomplish a different task. Therefore, when you press Ctrl-A, the terminal program does not send the ASCII code one to bash, and so bash never receives it. That's why bash isn't moving your cursor.
One solution, and quite possibly the very best solution, is likely in the terminal program's settings (which a comment, and another answer on this page, already address).
However, there is likely another easily available work-around. That is: Hold down the Alt key, and press the number one on the numpad. (If you are using a laptop, sometimes the numpad numbers are unavailable unless you also hold down a key labelled "Fn".) After pressing and releasing the num pad's number one, release the Alt key.
When you release the Alt key, the result is likely to send the specified ASCII code (ASCII code one) to the terminal program.
Yet another approach, although admittedly much more complicated, may be to somehow get ASCII code one into your clipboard, and then have your terminal program paste the contents of your clipboard (which can often be done by using Ctrl-V).
It should be noted that ^A doing select-all is consistent with other Windows (non-console) apps. – Jeff Zeitlin – 2018-07-16T19:10:54.950
@JeffZeitlin OK, so "right action" may have been subjective... hence the double quotes :-) – David – 2018-07-16T19:22:37.773