that pretty well covers it, other general benefits are that primes are typically smaller in size (though there are some massively large primes & many high quality primes have large glass that makes them larger than some zoom lenses!!!), which is useful when mobile or traveling. as well the simplicity of a single focal length, it does not really make for more considered composition though, rather a prime lens limits the options for composing that a zoom lens would give (with a zoom lens, the added dimension of deciding what focal length to use may slow you down)...
it really depends on need, yes primes are said to be sharper and can have larger apertures for low-light shooting/shallow depth of field photos but those things can achieved by higher ISOs and post processing, generally people rock with zooms for the convenience to change focal lengths fast. i think primes give a certain consistency/expectation to the experience of practicing photography, and to me are a bit easier/more fun to shoot with because once you get used to a focal length you kind of get an idea of what an image could be before making the picture...whereas with a zoom it may be a little more difficult...