On the topic "20k€ equity is nothing": There are very clear jumps in the interest rate level. More than 100%, because not even the additional purchase costs are covered - this results in really high penalty surcharges. 100%, purchase additional costs covered: already better. 95% loan-to-value: here there is another good discount. You can find, for example, all the conditions at ING on their website (except over 100%), so you can see what differences this makes. There are further tiers, but the jumps are smaller there. In that range, saving "pays off" the most. That means, at least the purchase additional costs should be provided by the borrower. If they have another roughly 20k€, they can at least break the 95% threshold, and that should also be the goal. In this respect, saving in this area makes sense despite rising prices, because the interest rate changes are greatest in this area.
On the other hand: my ex once wanted to finance an extension to his house; it was about a calculated (= professionally estimated, but without concrete offers) sum of 120k€. He had 10k€ equity and wanted to keep the loan absolutely low. The bank strongly advised him to keep this amount as a buffer, because with free tendering, the calculation might not be exact to the euro. But he absolutely didn’t want that.
The whole process was strange, I thought (just by the way), everything went through the regular payroll account. The bank transferred certain tranches, he paid invoices, then submitted them as proof that he had used the money properly. Then starting again. I don’t think he had an overview of what else from his salary “incidentally” flowed into the construction during the building phase. He was permanently "broke" at that time, I remember he once found an envelope from his father with 100€ in it (from his birthday or something) and was happy that he could now pay for the canteen meal until the next pay.
Result: The money was finally spent, then there was another invoice for 13k€. The family stepped in then. I thought back then, "he probably should have listened to the bank after all." Big difference: there were no additional purchase costs here except for the entry of the land charge, there was very high security because the land and existing house already belonged to him and thus the loan-to-value ratio was excellent.
For those who are not so comfortable, they should plan the buffer into the loan from the beginning. And then at least save the additional costs and 5% equity.