Could a material breach of contract be used to get an arbitration agreement thrown out? If so, how significant of a contractual breach would be required. I’m looking for a hypothetical example or even better some case law if it exists.
Under U.S. law, the answer is no.
You can defeat an arbitration agreement on the grounds that the contract containing an arbitration agreement was not entered into in the first place (e.g. the person who signed the agreement was not you but was instead another person whose name is also John Lee who also lives in Los Angeles, but is ten years older than you; or the person who agreed to the arbitration agreement was a two year year old randomly pushing buttons on a smart phone).
But, if a contract was formed, to the extent that the arbitration agreement so provides, all breaches of the contract, no matter how material, are subject to arbitration. Generally, this will be true, even if the contract was a contract of adhesion (e.g. a clickwrap contract).
The law outside the United States may differ materially from this rule.
In Australia, no.
An arbitration agreement in a contract is severable I.e. it is an independent contract and remains valid (assuming it is valid) even if the primary contract is void or terminated.
Also, breaching a contract does not end it - it merely gives the other party options for legal redress.