They're so ungodly heavy because of the batteries that they handle like barges. They need giant tyres and so much ESC and software control because these things weigh almost 2000kg or more. You can try and work around it but there's only so much that can be done to make 2000kg take a corner.
Looking at where sports cars will be in 10 years with ICEs being regulated out of existence makes me very sad because it seems like we're about to see the death of the lightweight sports car.
Under 1000kg for a reasonable price probably means building your own electrified exocar.
The weight of the batteries isn't going anywhere anytime soon. I expect car makers will prioritize range. I think the engineering to make an EV truly light like an old Integra Type R (1100kg) will be obscenely expensive and sacrifice so much on practicality it just won't be a viable product as a road car.
The car would be so compromised to be that light nobody will make the car, at least at an affordable price. You'll end up with a limited range, limited power, uncomfortable car for a price way out of line with what you're getting.
I think you could make a ~150kw-180kw EV pretty light, but considering the ongoing power pissing contest in modern cars I'm not sure how well it would market test.
So I expect the market will stick to heavy cars with big power because it's easier to build and easier to sell.