Answer

There is no reason it should not. "We are witnesses" substitutes for a requirement of testimony, so it can also substitute for a requirement of speech, so long as the speech is required only in order to convey the information. If the requirement is formal—that one must actually say it—then mere thought perhaps would not suffice. Thus, for example, regarding stipulations: something that is in his heart and in the heart of every person does not need to be stated explicitly (see Tosafot on Kiddushin 49 and many other places).
However, regarding conversion, the situation has changed in recent generations. In the past, when someone came to convert, there was a presumption ("we are witnesses") that he probably really believed and wanted to join us for proper reasons. And even if he was doing so because he wanted to marry a woman, in the past it was clear that he would have to join a community that observed the commandments. All Jews used to be like that. Today, however, people have an interest in joining—for example, in order to receive immigrant benefits and eligibility under the Law of Return. Therefore it is clear that today there is no such presumption. That does not mean that everyone comes with improper motives, but there is no longer a presumption that their motives are proper.
You can search here on the site for my articles "The Gates of Conversion — Violence and Good Intentions," etc.