Swivels are more than just a convenience for me, and I still don't think that they work as intended, probably partly because we cover one side with a nice lead clip or rubber sleeve, to stop tangles, but also because I don't think that line as it compresses and twists will exert enough 'pressure' to untwist. I think a swivel needs a fair amount of force to make it swivel.
One thing that they are very handy for is attaching the hooklink to the mainline. The eye is large enough to pass the mainline (or hooklink) through twice to create a very safe strong knot. I don't like joining lengths of line together, (mono leader or braided (leadcore for those that have to ) a knot creates a weak spot; so joining a hooklink with a knot to mainline is a worry for me, although I do it when I'm small fish fishing with fine lines
One very big advantage that swivels have in that join is regularly changing rigs, you automatically cut away a (short) length of the mainline, this piece of mainline just happens to be the section most likely to be damaged anyway, the lead rubbing over it, the run ring running up and down it, all will weaken that last bit especially, plus with the lead, it is the section that gets most force exerted on it during casting, causing stretch, and weakened line.