The reason the storage facility puts a limit on how much of a value you are allowed to put in storage (usually around $5000) is twofold. The first one is obvious as the OP has stated. They can "they can weasel out of responsibility by stating you violated the contract by storing high value items in the unit."
But let me explain this- the price of the unit is based on the size of the unit. Not the content. But the rental price remains fixed. Hypothetical example- one 10x10 unit is full of Hefty bags filled with dirty underwear and used shoes and such. The guy next to him rents a 10x10 unit for exactly the same price, but he loads it with gold bars. The rent is not based on the value of the stored items. By rights, shouldn't the renter storing gold bars be paying significantly more than the guy with the garbage bags?
Storage facilities are not in the business of covering the value of the contents. They don't temporarily "own" the contents while in storage. That would be known as bailment.
I know this won't be a popular post, but it is truthful and explains the limit situation.