Is my not dealing with my sister, who married عرفيًا (customarily/unofficially) without a guardian (wali) or public declaration, considered cutting off family ties, or is she the one who is cutting them off? Am I sinful for abandoning her, or must I prevent her by force?

Announcing the marriage (ishhar) is commendable but not a condition for its validity. The contract is valid with the presence of a guardian (wali), witnesses, offer (ijab), and acceptance (qabul). What the sister did by violating the Sharia in marrying without a guardian or witnesses makes her marriage invalid according to the majority of scholars, and valid according to the Hanafis. As for marriage without witnesses, it is not a marriage by the agreement of jurists. If the defect is only in the lack of announcement, then the matter is minor. However, if there is no guardian, the contract must be rectified. Cutting ties of kinship is not a solution; rather, she must maintain them, hoping that she will return to the path of Sharia.