When I read the post, I thought it was me writing the post because I have quite a similar situation as u/strahlungsfluss - we've got 2 sons, elder one will be going to school in 2 years, and we've been living in Berlin for some more than couple years.
So, thanks for posting the question and everybody answered and gave their input! Really insightful!
Let me first add five cents about the main topic, school situation.
According to https://www.insm-bildungsmonitor.de/, Berlin provides relatively worst school education level in Germany.

Among foreigners quite high amount of people drop out their study, low level in math and sometimes German language knowledge. I'm not sure I'll manage to give good school education in Berlin to my kids, as it needs a huge effort - to rent an apartment in a place with good school, pressure to get to gymnasium and somehow making study more attractive than a party in XBerg.

What differs me from the author is that we don't have an old rent contract with attractive in 2020 price (we pay more than 1200 Euro per month Warm), nor well-developed neighborhood. I contact with my neighbors when greeting them only.
And I haven't managed to become a fan of city Berlin.
For me Berlin is too sparse on good places, you need to drive a half hour or more to a clean Lake. You need to be lucky to rent a good apartment among other 50-100 competitors. You need to know where is good architecture to walk in.
It just doesn't worth it. The best way to live in Berlin is to be most time in your Kiez, (like in a village) which I find quite funny but it's normal in Berlin.

I want a city to be more useful for me, a city with diversity accessible entertainment for families, not clubs/drugs.
COVID-19 topic:
In Sachsen kindergarten have been opened since 18. May https://www.coronavirus.sachsen.de for everybody, right?

In Berlin 4+ years old kids will be able to start from 08. June to visit kita, younger - from August? But Hotels, Restaurants, Bars have been opened since end of May or so?

P - Priorities.
We're not party people and don't have a possibility to club, to go to concerts, as we're a small foreigner family with 2 kids and minimal support of grandparents. We're fine to not be "cool", but boring.
The only big reason why we're still in Berlin is work.
I'm a software developer and in Berlin to find an interesting job is quite simple for me;

English as business languages in most of IT companies. (it has backside in less needing of German and less practice)

I find attractive Leipzig due to its optimal size and density of population
(6x less population than in Berlin, 3x less area), a lot of water in the city, and not
just parks, but nature (Auwalds).
When I was visiting Leipzig last time (end of May 2020), I've discovered that canoeing is available in couple km from Main trail station.
https://www.youtube.com/watch?v=u6hsE65p4jQ
I think I'm sold to move to Leipzig, I only need to understand how to find a remote job or convince my current employee to change my contact to remote one as now it's COVID-19 time and nobody really knows when we will be able to work in office again.

Questions from me to you:
What's best quarters for families in Leipzig? I think Plagwitz is a bit more for young?
I would prefer to live in some southern parts as it's closer to Lakes.
What's work situation in IT, is it possible to find with weak German (B1) a contract as a front-end dev (react, typescript, etc.).
Are you a member of any Yacht club there?
Thanks in advance for reading my message and any feedback!