Hi - I didn't and couldn't say "nobody goes" as it's been over a decade since I lived in Istanbul. I can only say that I have never been to a beach while living in Istanbul and don't know anyone who went to beaches on the Black Sea coast. I've been to the islands and that was fun, although I can't say I saw great beaches there, either. At least back in those days, it was a bit boat-centred around the islands.
I guess it depends on where you live in Istanbul, as well: It would have probably taken me less time and effort to fly South to a proper beach destination than drive for hours to what passes for a beach on the Black Sea coast. And that is what I often did. I have friends who just booked flights for every weekend of summer, leaving Istanbul on Friday night and coming back on Sunday night.
Re Çesme/Alaçati - It is a place I know well, but would not recommend OP because she wants to be able to visit historical sites during her holiday and there just isn't anything of that sort around there. Ephesus, Pergamon, etc are extremely hot under the summer sun and even getting there will be a problem with two kids in tow - yes, she can rent a car but driving in Turkey can be very scary even when you know the way, and if there is anything worse than a day out with kids that involves taking multiple busses over many hours in a foreign country where you don't speak the language, frankly I don't want to know about it 
How is life in Istanbul these days? Traffic was already unbearable in my day when population was 12 million. I hear it's now 14 million. I hope you are enjoying it.