They can claim privilege for any evidence. All it means is that whatever it is it cannot be introduced as evidence at trial. For example if a third party debt collector sues Jim for an old credit card debt, and Jim challenges the claim and it goes to the discovery process, Jim would ask for the contract and bill of sale for the alleged debt from the original credit card company TO the third party debt company. If they admit to having the contract, but refuse to share it due to privilege, then the debt collector can't prove they own the debt, thus they can't show standing to sue Jim, case dismissed. The contract would end the relationship between the original creditor to the buyer, and no more documents will be supplied. So even if standing is established, the amount of money claimed could be challenged. If they don't have original documents, how do they rove how much is owed? Also these contracts would then be public record and the debt collectors don't want them available to the public for review and defense.
But if there is evidence in text messages that are not part of the lawyer/client interaction, and they claim privilege, that could be challenged and reviewed by the court. This is part of the ethics. In Jim's case the privilege claim helped him, so he won't challenge it. But in the Jones case the text messages showed Jones had consciousness of guilt and that helped the defense. So there is an ethical obligation to share the messages. If there were attorney/client messages they could be redacted. You can't withhold all texts if some are relevant to the case.