Broadly speaking, the repeater IDing itself is a good sign... It shouldn't do so if you had the offset or the PL tone wrong, and you wouldn't hear it at all if you had the recieve frequency wrong. It almost sounds like you are modulating a carrier, but aren't getting audio from the mic to the modulator... I had a similar problem with a Yaesu FT-8800 once; I replaced the microphone, and normality was restored to the universe... That's pure guess, though... Such a problem wouldn't manifest with a handheld, for example, not unless the built in mic was somehow damaged or defective...

You didn't say what radio it is... Does it have a microphone sensitivity setting, and is that setting turned all the way down? That would effectively mute your audio... Although, few FM rigs have such a setting, so it's unlikely...

Are you using a headset? That might have a bad wire or something... Alternatively, you might try a speaker/mic, as that would bypass the built in mic.

I'm otherwise at a bit of a loss... It would be diagnostically interesting to hear yourself over another radio, see what the problem actually is... You have the owners permission, see if he will help you troubleshoot.