Allan, as promised, I checked some of my resources and can confirm that the wooden ships' boats of the U.S.N. did have lifting rings. Bronze castings, "T"-shaped in section, were bolted to either end of the top of the keel (or keelson, as the case may have been) with the "leg" of the "T" pointing upward and the flat of it running fore and aft. A large bow shackle was permanently fastened with its pin running through a hole in the "T"-sectioned casting's upright leg.
The U.S.N., at least, did (and probably still does) have a specific drill for launching ship's boats using these lifting rings.
By all indications, this is also the same drill used by whaling ships. The danger of the boat "flipping" in mid-air while hanging from davit falls by these rings was never entirely eliminated. In fact, in most accounts of ship sinkings when crew and passengers must take to the lifeboats relate that some lifeboats, for one reason or another, capsized before they reached the water, spilling their passengers into the sea.
The U.S.N. boat launching drill is described in the fourteenth edition (1950) of the Bluejackets' Manual, which has most likely changed very little in the last 250 years in any navy:
When handled from davits, a ... whale boat is suspended at the bow and stern by separate falls. Usually, the block close to the boat has attached to it an automatic hook known as the "Raymond releasing hook."
When the boat is not in the water, the weight of the boat prevents the hook from tumbling off. When the boat is water-borne, the point of the hook tumbles and frees the boat.
Prior to hoisting, a lanyard made fast to the point of the hook is rove through the boat shackles and used to facilitate attachment of the hook to the shackles.
LAUNCHING THE LIFEBOAT -- ... If they do not have a definite assignment to carry out, they seize the lifelines as a safety precaution and remain quiet. Boat lifelines are knotted lines secured to the span of the davits.
When all is ready, the gripes are tripped and cleared away. Frapping lines and traveling lizards running on jackstays are used to keep the boat from swinging.
Frapping lines are passed around the boat falls and handled from the deck. The traveling lizards are kept in hand in the boat after a turn has been taken around a thwart. The lizards must never be secured in the boat.
Boat crew men in the waist ... breast off the ship's side with boat hooks; ... The ends of the poles not against the ship must always be held above the gunale of the boat lest a sudden swing against the ship shove the boat hook ends into the boat and stave holes in its sides.
Member of the boat crew stand by the releasing gear, and one man stands by the sea-painter ready to slip it.
When all is in order, the man in charge commands, "Lower away together," and the men handling the falls on deck begin lowering the boat.
As the boat goes down, other men keep a strain on all frapping lines and lizards; and men in the waist of the boat steady it with boat hooks. Any slack on the sea-painter is taken in. The man in charge of the sea-painter pulls the toggle and holds the end in the boat by hand. Only one turn of the sea-painter is taken around the inboard side of the second thwart. Other men in the boat tend the falls to keep them clear and to keep the blocks from striking other members of the crew after the falls are let go. When a tumbler hook is used, these men grab the tumbler lanyard and are prepared to unhook the falls in case they do not unhook automatically. If the boat is not supplied with detaching apparatus, these same men release the falls by hand.
The boat must be lowered smartly, especially in rough weather. The falls must always be lowered together. ... When the boat is a short distance from the water, the boat officer or coxswain lets go the detaching apparatus or commands, "Let go." If the boat is not fitted with detaching apparatus, the boat officer or coxswain commands first, "Let go the after fall."; then, "Let go the forward fall," as soon as the boat is water-borne. The after fall is always let go first. Remember that the ship is moving ahead, if the forward fall is let go first, the pull will be on the after fall, swinging the boat around and perhaps capsizing it.
The moment when the boat is water-borne is the most dangerous, and expert boat handling is required. Lizards are cast off. all members of the boat crew ust remain quiet and carry out only specifically assigned duties. The coxswain gives the stern a sheer in with the rudder to get the bow out. The strain on the sea-painter sheers the boat clear of the side.
When the boat is clear of the side, the officer or coxswain commands, "Cast off the sea-painter." An axe or hatchet should be ready for use in case of a jam at this critical point.
As can be seen, launching a ship's boat from davits is something of a "controlled crash landing" and quite a hazardous undertaking which required trained seamanship.
Before davits were commonly seen and where they were not present, falls from the masts were used in the same fashion.
Anyway, thanks for presenting the occasion to take a deep dive into "lifting rings" and ship's boat launching.