how to make cron run once at specific time per day?
I think you've covered the options well in your question.  However, your combinaton of requirements - i.e.:


no poking of cron.php from crontab;
settings kept in database to be migrated along with the site ("done from within a module");
the management module must not be Elysia.


is a problem. This combination excludes all the possible answers. There is no solution that will satisfy all three requirements.

If you want to use a module, that module is Elysia cron.

You seem to think Elysia cron is overkill for this.  I happen to agree.  Besides, Elysia does not work too well when triggered by the built-in default poorman's cron.  You really want to use an external crontab poke together with Elysia to get the type of fine-grained control you seem to be after, with Elysia.

So while your first requirement rules out poking of cron.php from crontab, that is my suggested solution.

Actually, since the crontab entries for all your sites can be kept in a single file on a single server, I do not see how this set-up should be considered too difficult to maintain.