I don't find €3000 as a monthly payment that unrealistic given that salary. At the beginning of last year, we signed for a €2000 payment with a net income of €5500. Just like with you, that leaves just under €3500 to live on. We are currently trying it out since we already pay over €500 in interest and of course still have the €1600 rent. You can live well on that and even afford an all-inclusive vacation for €500 per person. Then it will just be Crete, Mallorca, Turkey from Check24 or something. Especially with the bonus payments you apparently still receive.
However, we don't have kids yet, and that will probably make things overall a bit tighter than for us, but it doesn't sound completely far-fetched. Keeping a good household budget and not spending money unnecessarily on “crap” is always wise anyway.
The big problem I see for you is the comparatively old house. We are currently building completely new on a 350 sqm plot, and the maintenance and ancillary costs will be as low as possible for a house for quite some time. Even if you manage with the €150,000 renovation, which can always increase, costs may of course come up with an older existing property. Ancillary costs will also be significantly higher. Especially installing a gas heating system right now can be a financial disaster. It's foreseeable that in the next 10–15 years there will be a heavy CO2 tax, and you will want to keep a new heating system for 20–25 years. So if new, then do it properly right away.
If you can clear up these concerns with a good expert and make a solid plan, it can definitely work. I don't see the problem with the payment, but the age of the property could become an issue.
Be aware, though, that such a large house with such a huge plot is also a lot of work. Once the kids are old enough, the whole family needs to be involved, otherwise you'll wear yourselves out.
It should also be mentioned that you are just as young as we are. The salary will definitely increase over the years, making things easier. For example, mine does annually as inflation compensation. That works well. From what I've read here, that is a collective agreement for you. Those also increase steadily if I am not mistaken.