Building near Leipzig, which budget / financing / which options?

My brief advice would be to come back here first (rental apartment), take jobs, and then the property.

Actually, we wanted an existing property, but we have been watching the market for a while and there is practically nothing available.

Saw it by chance today. In the mentioned commuter belt, there is a new bungalow. Can be found on the well-known portals or directly at Saalesparkasse.

Whichever way you decide, good luck!

So I assume that you are not seriously looking for a building plot IN Leipzig. I believe by now that is unrealistic in any reasonably sought-after major city. (either they don’t come on the market or the prices are astronomical)
In the extended Leipzig surrounding area (20km) it gets significantly more affordable. There you can get building land for under 150 euros/sqm.
Then you also don’t need a 620,000 euro budget.
Blah blah... all purely theoretical. As I said: the first step must be to start job searching in Leipzig and at least theoretically outline your future scope of action.

We want, as soon as everything is planned, to work on having a child and tend to come back around the time of birth.

Exclude child planning from the temporal/financial planning! Children come whenever they want. That can hardly be planned.
We were always warned that it could take years. Then for us it was one shot, one hit, and everything happened much faster than expected.

everything much faster than expected.

Or the other way around. And that can become very expensive...

Better stay there for another 1 - 2 years, child. You will never save as much money there (of course without the child) again in your life. We were there for a good 10 years and then simply paid cash for the cabin here in DE. The best of all solutions.