A ship with the weather gage can choose when and how to engage.
> This was technically inferior since the lower gun ports could often be underwater (see image) and because the downwind (leeward) position made it easier to flee if needed.
This is reductive to the point of error. The ports of the lower gun deck MAY be unable to be fired in very heavy seas, but that doesn't affect frigates, or the upper deck of a ship of the line.
Additionally, if a leeward ship attempts to flee from the windward ship, the leeward ship would risk exposing its stern to the windward ship's raking fire. The stern of a ship is the least armored, least armed and also contains the essential steering elements of a ship. A stern raking fire could pierce the hull and fly the entire length of the ship, causing tremendous damage, in addition to potentially crippling a ship's ability to steer.
Finally, the encouragement to engage with the enemy has an advantage the article omits - massive career incentives - it's a chance for British Navy lieutenants and commanders to earn promotion. Many a commander was made post after a successful engagement with the enemy and many a lieutenant was promoted to commander after a successful battle. Beyond glory, a lieutenants would make roughly half what a commander made, and a post captain could rely on additional pay based on seniority and ship. Since promotion to admiral was almost solely due to seniority on the post captains list, naval officers felt urgency to win promotion and to get on the list as soon as possible. An admiral took a share of any prizes won by vessels under his command and was the true way to gain wealth in the Navy.
Finally, Byng's case is an extreme outlier and relying on it to make arguments is dicey at best.