Can we use "${a:-b}" for variable assignment in bash, with "a" a command?

In zsh, you can do:

var=${$(cmd):-default}

But in any POSIX-like shell including bash, you can always do:

var=$(cmd)
var=${var:-default}

Or

: "${var:=default}"

You do lose the exit status, but you could save it in a variable in between the two steps:

var=$(cmd)
exit_status=$?
: "${var:=default}"

You can also do things like:

var=$(cmd) exit_status=$? || var=default

For $var to be set to default if cmd failed (regardless of whether it output something or not).

Note that if cmd output is non-empty but consists only of newline characters, then after var=$(cmd), $var will still be empty as command substitution strips all trailing newline characters. bash also strips all NUL characters. zsh preserves them, some other shells strip everything after the first NUL character. yash chokes on sequences of bytes not forming valid characters in the locale.

For $var to be set to default only if cmd's output is non-empty, with GNU grep (or any implementation that can cope with (and not modify) non-text input, and doesn't ignore non-delimited lines), you can do:

var=$(cmd | grep '^') || var=default

where grep '^' returns true if it finds the beginning of at least one line in its input (and passes its whole input undisturbed to its output).

In that case, cmd's exit status is lost. $var could end up being empty if cmd outputs only NULs or newlines.