Does it make sense to have an additional insurance policy on the flat in addition to the security deposit I give to the landlord?

Absolutely yes. The security deposit is at the end of the day a small amount of money able to cover small damages of the flat, like a broken glass, window, door, furniture or even an entire kitchen at maximum but it's definitely not up to repairing big damages that you could make.

Example: you leave the sink of the bathtub open and you destroy your flat and the one downstairs... the damage could be easily up to 40-50k EUR, the landlord would immediately use your deposit but would also sue you for the remanining amounts due. 

An isurance policy, usually very cheap, in the order of 10 EUR a month, would prevent this. In Italy we usually suggest our clients to go to Poste Italiane and ask for the "Assicurazione del Conduttore". 

SPend this money, you will not regret it.