Looking, it turns out on my system there is an envsubst command which is part of the gettext-base package.
So, this makes it easy:
envsubst < "source.txt" > "destination.txt"
Note if you want to use the same file for both, you’ll have to use something like moreutil’s sponge, as suggested by Johnny Utahh: envsubst < "source.txt" | sponge "source.txt". (Because the shell redirect will otherwise empty the file before its read.)