Detached house on a hillside, approx. 220 m², 2.5 floors, gable roof - ideas?

Regarding the house shape, it just looks like a gable roof house. You can't change much there, except for a completely different house style.

For the stairs, I could imagine a straight staircase given the size.

Be sure to get offers including all earthworks. Your budget is tight and without personal contribution it won't work.
Also plan generously for the outdoor area. Here excavation, there a wall, there filling.
Not every excavation can be filled. We have to haul everything away, more stones than soil.

Children live in the garden. You run yourselves ragged. Had the pleasure for 2 years. No way. Part of the garden fenced off and childproof, nothing can happen there. Balcony railings invite climbing.

Thank you very much for your constructive feedback. As mentioned: we have weighed the pros and cons of placing the living space in the basement quite thoroughly against each other. According to our planner, unfortunately the straight staircase does not work.

It is planned with a relatively high amount of own effort. Nevertheless, there is still a buffer in case it should end up being somewhat more expensive.

...Unfortunately, the straight staircase does not work according to our planner....

"Unfortunately"? Or not important to you?...If the former, I would reconsider the planner...

The straight staircase was a topic at the beginning of the planning and was not possible at that time. However, I do not remember the reason right now. Since several things have changed during the planning process, it might be possible now. We simply did not address the topic again afterwards, so thank you very much for the hint!!!

Straight stairs work for the area.

Still, add up all the prices together. Best are proper offers without own contribution; you are about 100,000 euros over budget. It is hardly possible to recover that with own effort.

Take a look here, the costs are roughly broken down. Some depend on the size or the property, others are size-independent. Use it as a guideline.