Buying a house with cash, is it sensible? No trouble with the bank, installments, etc.

It depends on how the money is invested. Normal savings methods just don’t bring interest gains. Everything speculative would be too risky for me. And if it's invested that way, then the question is to what extent the bank accepts that as collateral.

Of course, if the money is in an instant access savings account, it looks different. But in my opinion, it also makes no sense to finance the house then.

Apartment for rent as a capital investment? I find that nonsensical, as the apartment is paid for in cash while the home is financed and the interest on the debt cannot be deducted for tax purposes. Then there are the risks that a landlord has to bear (e.g. rent default).

That was not meant that way.

However, one can finance their own house, for example, with 60%.
With the remaining money, the required equity for the purchase of a multi-family house is raised.
The own single-family house is then paid off quickly, while the multi-family house is repaid very slowly, since the interest is tax deductible.
Of course, the multi-family house should generate a better return than the costs for the single-family house financing.
Just look up "leverage effect" on Wikipedia under the section capital structure.