My car has low profile tires-they offer good handling, but they have a lot of disadvantages-if you so much as bump a curb, you lose pressure (I am always filling my tires).
Suppose I could get solid (or foam filled) tires?
The ride would be harsher (but on good smooth roads-would I notice any difference)?
Plus, no flats or blowouts.
Have solid tire ever been offered for passenger cars?
Go to Walmart.
Get in a shopping cart.
Have your mother push you around in the parking lot.
Now you know what it would feel like.
I think you’re missing some shocks and a suspension there.
Using that much rubber and other solid materials would probably be too expensive and add a great deal of weight to the wheels.
A shopping cart has no suspension, so yes, it would have a terrible ride.
A closer analogy for solid tires on a good smooth road would be a train. Solid steel wheels on smooth rails and a good suspension between them and you.
Solid tires tend to lose chunks of themselves on rough terrain as there’s no resiliency or cushion to them. I’ve seen forklifts with solid rubber tires that last nearly forever if they stay inside on smooth concrete floors, but as soon as someone ventures outside, they get all chewed up when they hit potholes, rocks and all the other stuff that’s on a typical piece of pavement.
What is this “good smooth road” you speak of? I’ve seen pictures of them on TV, but they sure don’t exist around here.
You’d lose grip, as your contact patch would shrink to almost nothing. You’d slide all over the place while you rattled your teeth out.
Lots of cars are sold with run-flats that have ultra-stiff sidewalls. Sadly, they’re often equipped on sports cars, which have small trunks and can benefit from ditching the spare tire. It’s generally regarded as one of the worst things to happen to high-end sports cars since the automatic transmission.
While a car certainly has a better suspension than a shopping cart, a car also weighs a lot more and travels a lot faster. The ride would be horrible on solid tires.
Michelin created the “tweel” a few years back, which is an airless tire. It’s a neat design (google it to see what I mean) but it suffered from a harsh ride as well as vibration and noise problems. I can’t imagine it would handle very well either, especially for someone who likes the performance of low profile tires.
For those who don’t want to get stranded by a flat or a blowout, there are plenty of “run flat” designs out there. These don’t run as well if they are punctured, but you can drive on them well enough to get you home or to the service station.
I live in Pennsylvania. Around here, a good smooth road is one you don’t need a 4 wheel drive truck to drive on it. 
I think solid tires would be completely shredded before you even got to the end of the road I live on.
That’s why fork truck have indoor wheels and outdoor wheels. Indoor wheels are made of a type of polyurethane - not rubber. And while they have great traction and long life on smooth concrete, they don’t last outside (nor were they designed to).
Outdoor wheels are solid rubber and will last outside much longer than polyurethane wheels - but not nearly as long inside.
Anyone wanna guess what I do for a living?
As for the OP, I think your car would ride just fine on solid rubber - especially compared to the low profiles you’re running. But, solid rubber would weigh more, cost more and not last nearly as long due to chunking out.