The US had a complicated relationship between weapons testing and publicity. Its earliest test, Trinity, was of course done in Top Secret, and footage was only released after the bombing of Hiroshima so that people (Americans and otherwise) could understand what the atomic bomb was (whereas footage of the Hiroshima bombing itself was not). Its first postwar nuclear test series, Operation Crossroads, was open to guests of many nations, including the USSR. But after that it became much more withdrawn about its efforts for many years, because it became more concerned with developmental testing (testing new designs) and what those tests revealed said much about the state of the US arsenal. So while they announced the existence of nuclear weapons test series (they are hard to "hide"), they released very few photographs or information about the tests themselves, such as when they were being detonated and their yields. They attempted to keep the results of Operation Ivy (1952), the first H-bomb test, a total secret (to limited success), but after the first H-bomb fallout accident in 1954 (Castle Bravo), they released a lot of the Ivy Mike footage (because they weren't willing to yet release anything about the Bravo footage).

By the 1950s they were also testing in Nevada, and those were visible from nearby cities like Las Vegas, and there were even an area set aside near the test site where news organizations could film the tests from afar ("News Knob"). The information released about these tests could range from very minimal to very extensive depending on the specific goal of the publicity; the famous test of the Genie missile above the heads of several military men (and a camera man) was highly publicized because it was about showing the United States citizens that high-altitude, low-yield weapons used for defensive purposes could be done relatively safely.

Over the course of the 1950s, they did occasionally decide to release certain clips of nuclear tests from earlier periods. These releases were generally highly-selective and for that reason, if you track these kinds of things, you see the same footage over and over again (despite there being much more footage than this created). So some of the footage was released quite widely, but most of it was not. Such was generally in line with the all-or-nothing the US approach to nuclear secrecy; they either decided something was safe enough to give to everyone, or they kept it under very tight wraps.

US testing was certainly seen by those who did it as a form of deterrence in and of itself (a vigorous testing program was seen as providing credibility, aside from the benefits it gave to stockpile development), but I think that assuming they released the footage as part of a concerted deterrence campaign against the USSR is probably giving the people in charge of releasing it (the Atomic Energy Commission) too much credit. They did not generally think or act in big, strategic, geopolitical ways. They made nukes and tested nukes and, if they thought it was safe to do so, released stuff. That was part of their legal charter and part of their normal operating procedures.

As for Stranglove, the tests shown in it are indicative of what I mean by some things being released extensively if redundantly. They are really only showing 5 or so different tests, cut up to look like many more explosions than it is (Crossroads Baker, which was a very distinctive-looking underwater test, makes up nearly half of it, from different times of its detonation).