Acceleration is better than nothing. But schools specifically for high-IQ kids would be even better -- there you could be with kids your age who would simultaneously be your intellectual peers.
Unfortunately this idea is politically unacceptable in most places. People don't want to help form an "elite" (especially if they or their kids couldn't be a part of it).
Maybe the concept would be more accepted if the nation didn't have a legacy of intentionally depriving tens of millions of citizens of education based on the color of their skin, and if elites didn't grow up to hoard the nation's wealth for themselves and disregard the needs of the majority. Trustworthiness matters.
Nothing like sacrificing today's academically-elite kids of non-elite parents for yesterday's elites' actions. (Today's kids of elite parents will do fine either way, of course.)
The exact details depend on a country, but generally there are two kinds of problems:
Financial -- In some countries, the public schools are fully financed from taxes, but private schools only get a fraction of that money, and the rest needs to be paid by parents. So even a school that has the same expenses and does not try to generate profit, could still be too expensive for many parents.
Curricular -- In some countries, private school doesn't necessarily mean that you are allowed to teach things differently than the public schools. You may spend the extra money to have larger classrooms or larger playground, fewer kids in the classroom, pay the teachers higher salary (ha ha, never happens), buy fancy uniforms for the kids, or just put the money in your pocket as a profit. But you are still required to teach the curriculum dumbed down for the average kids. So, yes, the kids could get smarter peers, but they still couldn't leverage it into getting better education.
The thing is that most people would agree that all kids should get "equal service" for taxpayer money, and the rich kids should be allowed to pay for "extra service" if they want to. However, you have two different interpretation of what "equal service" means. It can either mean that each child gets education proportional to their abilities, or it can mean that each child gets education proportional to the average child's abilities.
It's not like teaching gifted kids more advanced curriculum is inherently more expensive. You could use the same building, have the same amount of kids in the classroom, pay teachers the same salary... the only difference would be progressing 10 pages of the textbook in time the average school progresses 5 pages. But some people still insist that it's not fair for gifted children to get more education on taxpayer dime, even if the costs are exactly the same.
Unfortunately this idea is politically unacceptable in most places. People don't want to help form an "elite" (especially if they or their kids couldn't be a part of it).