The Stratocaster actually wasn't that popular in the '60s. If you watch the Woodstock movie, for example, you will see more Gibson guitars of all marques, and Fender basses in the main. In the '60s the Strat was almost entirely synonymous with surf rock and instrumental music, and couldn't reliably drive the front end of a tube amp in the same way that Gibson instruments could, which was the sound that a lot of guitarists were chasing in the latter half of the '60s. Hendrix is the exception, of course, but there is an almost super-human element to how he chased such a rich, thick tone from a Strat.
You mention some guitarists, but they probably all have different reasons for adopting the Strat, ultimately. Stevie Ray Vaughn didn't really surface until the '80s, so should probably be excluded from the list. Jimi Hendrix started using them when he moved to London in 1966, perhaps because in part he found the whammy bar to be a useful and expressive tool. It might just be the first guitar that he was given when he arrived, and quickly he became iconic for playing them reversed? He used Strats for stage work, but is photographed using many other instruments in the studio.
Jeff Beck used a Strat with the Jeff Beck group in the late '60s, but made far greater use of a couple of Les Pauls during the period. He used a Strat more from the mid-'70s onwards as it complimented his playing style more as he stopped using a pick and used the whammy as an expressive tool as well.
Eric Clapton possibly used one as a direct influence of Robbie Robertson of The Band. He was also moving away from Gibsons, which he used exclusively during the Cream era (with the exception of a Fender XII on 'Dance the Night Away'). Clapton took press reviews of his Cream period badly, with some claiming he was little more than taking a 'copy and paste' approach to earlier blues guitarist's licks and riffs. Perhaps for him the thinner, wirier Fender tone was a clear distinction from the meaty Gibson roar he had become synonymous with.
Generally speaking the Strat is ergonomic and quite comfortable to play for long periods, as long as you get on with the neck. Of the most popular guitars of the '60s, the Strat is more ergonomic than a Telecaster, lighter than a Les Paul and balanced more towards the body than the notoriously neck-heavy Gibson SG. Strats are also harder wearing. Watch the Rolling Stones' Gimmie Shelter film and you will see Gibsons haphazardly stacked against and resting on top of amps. Touring guitarists perhaps didn't baby their instruments so much back then, so Gibson headstocks got broken quite frequently. Stratocasters are modular, so broken parts can be quickly unbolted and replaced. Hendrix favoured some Strats and carried a cache of mostly sunburst, rosewood necked 'beater' Strats he would close a set with. There are photos and video evidence that suggests he typically threw the Strats into his amps to round off a concert, which would quickly break a Gibson.