canada*
Solicitor-client privilege is the client's to waive.
In the case that a lawyer inadvertently discloses information that would be protected by solicitor-client privilege, that is not an automatic waiver of the privilege. See R. v. Ward, 2016 ONCA 568 at para. 35:
Inadvertent disclosure does not necessarily mean that privilege has been waived. While waiver of solicitor-client privilege can be express or implied, whether privilege has been waived by inadvertent disclosure is a fact-specific inquiry, which may include consideration of the following factors:
• The way in which the documents came to be released;
• Whether there was a prompt attempt to retrieve the documents after the disclosure was discovered;
• The timing of the discovery of the disclosure;
• The timing of the application;
• The number and nature of the third parties who have become
aware of the documents;
• Whether maintenance of the privilege will create an actual or
perceived unfairness to the opposing party; and
• The impact on the fairness, both actual or perceived, of the
processes of the court.
(These or similar factors also apply whether when disclosure is due to the client's inadvertence.)
You ask:
Is something implying "even their own lawyers don't believe their own arguments" not something you're allowed to point out in court?
It would be irrelevant, unless the case turns on a defence based on legal advice. If the substantive legal argument is weak, it is sufficient to just point that out and make your own legal argument. Nothing is added, legally, by pointing out that the other side believes their argument to be weak.
In court, it is extremely rare that a lawyer's belief is relevant to the issue. In almost all circumstances, all that matters is the evidence about material facts, and the lawyer's submissions about the law. The lawyer is only stating their argument, not their belief. A typical phrase used by a lawyer putting a legal argument to the court is, "I submit that..."
E.g. if the question before the court is whether something is a thing of value, all that matters is what the legal test is, and what the material facts are. Whether the lawyer advised the client that it was a thing of value or not does not change whether it is in fact a thing of value.
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