Your post raises more questions than answers can provide.
Is the snow removal promise in your lease? Or is it that you know the landlord used to have a contract with a third party to shovel snow? Where was the snow allowed to accumulate? What damages do you have that you can prove?
While there are lots a case opinions about the concerns of removal of natural and unnatural accumulations of snow, until the above (and many more) questions are addressed, far too many variables exist in the short post you wrote for any further observation by me.
Your situation is too involved for an online answer.
Using the written documentation you have is a good place to begin. Contact a local lawyer - many may give you a short free consultation - to discuss your specifics.
You might find my Legal Guide helpful "How to Choose A Lawyer For You"
You might find my Legal Guide helpful " What Do I Tell My Lawyer"
No one can know what the record is in the case because online we cannot see your lease documents. You need a lawyer. Check with a lawyer in your locale to discuss more of the details.
Good luck to you.
NOTE: This observation is made available by the lawyer for educational purposes only. This observation is not like a communication with a lawyer with whom you have an attorney-client relationship along with all the privileges that relationship provides.