Sixth grade is a less common entry point for privates than 9th. It definitely happens, but if you go public for K, you need to be open to the fairly high likelihood you will stay public for middle school. Most of the areas that have good Gen Ed public elementary schools also have good middle schools, though you don't always have full control over the process (i.e. random lotteries that don't factor in academic performance). But if you are just looking at elementary now, that process could change several times between now and when you are considering middle schools.
My 12yo attends a zoned District 2 middle school with a screened program, something I think they added largely to console families with bad lottery numbers, and while it's not exactly *super* rigorous - 2/3 of the 6th grade is in it and the remaining 1/3 has the chance to test into it at the end of the year - it does at least prevent the scenario where one or two struggling kids occupy an inordinate portion of the teacher's time. And the kids are pretty smart, I think like 1/4 of her class qualified to take the Hunter test (though none got in), compared to roughly 5% of kids citywide.
Teachers / facilities are a mixed bag but I'd say 3 out of her 4 core teachers were solid and the theater program is very big and ambitious - 2 full musicals a year, more than most private middle schools manage, and several 8th graders are going to LaGuardia for drama next year.
But basically, the worst-case scenario for public middle school in a nice district is not bad, certainly not bad enough to be worth paying for 6 years of unnecessary private school to avoid it.