The images were hits in newsreels, magazines, and all attempts at debunking any and all reports of unidentified flying objects. Long before there was any such project envisioned military and governments claimed they had "flying saucers" and they were really just radical experimental designs.
In late 1956 and 1957 the USAF entered a contract with the Avro Aircraft Ltd. company to design and construct a working prototype of an aircraft able to transcend some specific or particular issues. The AF was aware of the limitations of existing traditional landing and take-off situations - long runaway, wider runways, needing thicker and thicker substructure construction for the increased weight and loads of aircraft. They liked the idea of vertical take-offs and landing and explored numerous designs in early helicopter and vertical takeoff craft.
In February of 1955 "Project Silver Bug" seemed a viable solution. In the report to the USAF of that date they explained that the resulting disc shaped craft was not influenced by science fiction or accounts of objects in the skies but from designers and technical types trying to answer the Air Force's specified need for "dispersed base operations." They needed a craft that could get in and get out without the long runway hindrance.
A result was PROJECT 1794 aka Project Silver Bug.
I do not believe these were the products of copying downed alien craft; they would have worked better if that were the case. Sadly, the end result seen in old films is a less than stellar demonstration of an object struggling to stay level at only three feet of altitude.