and no paperwork goes into subletting for such short durations
Why not? In the Netherlands, it is completely normal to set up a basic contract even for short stay subletting. There is no harm in asking for a simple contract you and the 'landlord' (i.e., current tenant) can sign. There are only benefits to having a piece of paper proving that you're allowed to stay there. Since Dutch law provides extensive protections for tenants, the contract does not need to be complicated (there's plenty of templates for your landlord to choose from online). Usually, the contract just states what you're renting (something like "a single room, and use of the common areas including the washing machine"), for how long (start and end date) and how much to pay each month to whom.
By the way, just because the university doesn't provide you with housing, does not mean you can't use housing aimed at foreign students. Most cities with a university have a building cooperation aimed at student housing which generally also provides short stay housing. There may also be other commercial alternatives like student hotels.