Is the outdated PHP mysql extension a requirement for WordPress 4.6?

WordPress will use MySQLi when it can or unless you tell it not to. Unfortunately if WP doesn’t see a mysqli_* function, it will assume that you want to use mysql_*.

There is some logic that will fall back to plain, old mysql_*, but, from your stack trace, that’s not what’s happening.

You don’t mention what OS you’re running on, but I’d guess that the php70-mysql package was not installed. Ubuntu and things like the ondrej/php PPA won’t include the various mysql drivers (pdo mysql and mysqli) by default.