Could they be harvested? Sure.
Could they be harvested in an economically profitable way? Probably not, for all the reasons you listed.
There's a lot of unknowns to this question — while the chemistry they mention certainly makes sense, there's lots of stuff about the atmosphere we don't know. A harvesting ship would need to find exactly what altitudes the diamonds formed, but haven't yet been crushed into the core of the planet. Remember, it's not like there's millions of tons of diamonds just sitting on the surface, there's an altitude range where they're formed and haven't yet been destroyed. This range appears after a lightning storm, so you'd need to send your harvester underneath the graphite 'cloud' at the right time-frame to harvest.
Pressures high enough to turn graphite into diamond means that any ship would need to be ridiculously strong to not also be crushed.
If we could get to the point technologically where asteroid mining is feasible, there might be more opportunity to harvest materials from the gas-giants.
The other trick is if you were able to harvest them effectively and in large quantities, selling tons of diamonds would likely destabilize the market, and probably cause a price-crash. On the flip side, 'Saturn Diamonds' might become a huge status symbol and command a high-enough price to actually make such a venture worthwhile.
From what we know, carbon is a very common element in the universe — there might be entire planets that are carbon-dominant instead of silicon-dominant, meaning thick-swaths of the planet's crust could be solid diamond.
Given that we've been able to create lab-grown diamonds on an industrial scale for decades, even with huge leaps in space-exploration technology, it is still going to be much easier to synthesize diamond than it will be to harvest or mine them from off-earth sources.