I've lived in Berlin for virtually my entire adult life, as has DH, and our DC were born here, logically enough. In terms of quality of life, I would absolutely recommend Germany over the UK. Costs are much lower we find, more culture and childcare is subsidised, there's less obsession with the free market at the expense of people. There are still no or only low university fees. In Berlin there are a number of state schools which offer German-English education. In short, our standard of living is much higher here than it would be in the UK, for the kind of things we find important.
In the East you'd have most chance of finding somewhere as a HCP out in the countryside, because there's is so much depopulation (are there any doctors left in Meck-Pom?) - and that's your problem, because if my family were mixed race it would be a cold day in hell before I'd go to live in certain parts of the East. Leipzig, hmm, I don't know. DD1 was in Rostock last weekend with her school on a trip, all of them were 15 to 17 years old, and they were attacked at a tram stop by a drunken racist who kicked the non-white people in the group (and it's an international school so ethnically very diverse) - the police were called, all very unpleasant. I was attacked by neo-Nazis in 1993 with friends in eastern Berlin, just for speaking English - things are very different there now, of course, but at the time I was worried enough when I got pregnant to move to the West part of the city. You'd be safe enough in Prenzlauer Berg now - perhaps that would be the best way of combining staying in the East with a safe environment for your DC. But the education in P berg is not the best, and there are no state Ger-Eng schools there, only private ones. Rents in P Berg have gone up massively recently due to gentrification (hence the need for a rent cap). You could get this:
www.immonet.de/angebot/25817656?drop=lp&exproot=3-zimmer-wohnung-mieten
in P Berg near Schönhauser Allee for 810.30 euro cold, 105 sq. m., Altbau
or this for 381.20 but it's a smaller Neubau:
www.immonet.de/angebot/25816079?drop=lp&exproot=3-zimmer-wohnung-mieten
We all got German citizenship last year, also partly because of worries about a potential Brexit (yes, Pippi, I'm sure in practice a solution would be found so we wouldn't get kicked out, but there are plenty of things which are more difficult for Swiss people living in Germany than for EU citizens), but mainly so we could vote and the kids would have all jobs open to them.
Child benefit is 184 euros per child! Yay! It's still not persuading people to have any more children. Another reason to live in Germany - lots of employment opportunities for your kids in years to come.