If it's the big wooden "persianas", it could be quite expensive to replace. And not at all the responsibility of the renter, if it was storm damage or any other kind of natural weathering. Even if the contract says everything is new. The strap to open and close is probably another story - they are indeed relatively inexpensive to replace and would fall under normal wear and tear. If they were already broken when you moved in three years ago, you wouldn't technically be responsible for that, although signing a new lease saying everything was new, that's going to be hard to argue.
I'd go ahead and replace the straps and tell them "no" to replacing the blinds themselves.
We used to live in a house in a closed neighborhood near Garin, outside the city a ways. When we left, the owner inspected the house and tried to convince us we had to replace all of his exterior eaves around the house, which were dry rotted. Hadn't been maintained in years, it was in bad shape when we rented the house. We didn't do it. Or in our recent apartment fire, the owners are trying to tell us it was our responsibility to replace everything, including kitchen cabinets that were falling apart previously and weren't anywhere near the fire, and even though the fire was apparently the fault of a bad electrical junction box near the stove (what a place to put a junction box anyway!). As well as electrical circuits in other parts of the apartment that hadn't worked for some time and they hadn't fixed.
You do have to careful about what you pay for here, many owners will try to get you to replace things that are not your responsibility. But the renter is, indeed, responsible for other normal wear and tear items in the apartment. It can sometimes be difficult to ascertain exactly what that should be.
