Honestly I've no idea why anyone buys progressive springs. I suspect not knowing any better plays a role.
As to why they exist, I suspect it's a bit like why all season tires exist, despite them delivering inferior results to the alternative.
They're a compromise that aren't great at anything they try to do, but they kinda give you some of both worlds.... just like All season tires work "ok" in the summer and winter, but come in 2nd or 3rd place compared to tires dedicated to working in those conditions.
Progressive springs are softer under light load, and firmer under hard load.
In theory this means a comfortable right when just cruising around, and enhanced performance when pushed.
And for folks who when you're honest about it spend 99.9 of their driving on smooth highways and smooth paved roads, and don't really push the car very hard, they seem to be fine and hey, "slammed" is sexy right? So for these people who just want a "drop" they can buy the springs and until they're on a bad road won't notice there's anything wrong at all.
But for folks who go over more significant bumps or poor roads, they suddenly notice progressive springs kinda suck because it impossible to properly match a shock to them since their rate changes.... so suddenly their F-sport springs with stock shocks are unpleasant over big bumps... and those stock shocks were out a bit sooner than they otherwise would too.
And if they take it to a track they notice the spring rate changes halfway through a turn, which can be an unpleasant surprise.
Mostly I think the F-sport springs (which are really just rebranded aftermarket lowering springs after all) exist because it was cheap/easy for Lexus to slap their name on em and offer a way to lower the car inexpensively.
I suspect if you actually ran a stock sport-suspension car at the track versus one with F-sport springs on it, the stock sport suspension car would turn in better times. And the stock one will certainly do better over larger road imperfections.
If I were ever looking to lower my car I'd do it with coilovers- even if I had no intention of changing heights on a regular basis- because I could get linear springs with custom rates and a shock designed to match it.
With progressive springs your shock will always be either overdamping or underdamping for one end of the spring rate or the other.
But again, all that said, on smooth roads, the progressive springs will give a softer ride than the stock sport springs will, because they spring rate is lower... probably not as soft as the stock non-sport springs though.
When pushed the F-sport springs won't be as predictable as the Sport springs, but they will be firmer than the stock non-sport ones.
Linear firm or linear soft will always be better at one of the two tasks, but progressive springs are "ok" at both. Linear either will be better on big bumps because of being able to dampen properly.
So, prog springs are a compromise to let you lower the car and usually be ok at most tasks. Not a compromise I want, but many others either do want it, or don't realize what the drawbacks are, or don't realize there's other (possibly better) options.