Yes, the boars of that era will have a lifting eye in the bow & stern of the boats--these would be hidden under the boat cover. Those eyes are inboard of the stem & stern of the boat--ideally, the sames spacing as the davits.
To use radial davits, you hoist the boat out of it's chocks. Then both davits are pivoted until the boat either the boat rubs a davit, or either bow or stern clears. A bit of padding is put between the davit and boat and it is heaved a bit as the davit is swung about (there is a socket in the davit for a bar to better pivot it). When one end of the boat is clear, the other davit is heaved around until the boat is entirely over the side hanging in its falls. Which is also probably a good spot to note that Revell chose to show the davit "winches" turned 90º from a useful direction--which was likely a choice made as a limitation of the casting process).
This process requires 6-10 stout lads per boats, which is why radial davits were replaced with patent davits for lifeboats in maritime service.
Please. please, do not leave the boat falls dangling over the side unless you are depicting either the immediate release or the collection of said boat. It's not merely a bit of "modeler's license"--it's more of a faux pas on the level of depicting a modern jet fighter in flight with its canopy open. The boat falls would be heaved in to at least deck level and lashed to the davits with marline
In all probability, the CS crew probably more regularly just used the tackle for hoisting things out of the holds, and just used a sling to pluck the jolly boat up and over the side.
When not in use, a boat would probably just be tied off from the stern by it's bow painter