Wouldn't Hebrew text instruct followers to sing unto God? If you are in the shower, naked, cleaning dirt off yourself, it seems to me that that wouldn't be appropriate . . .
Your proposal would make better sense if the rabbi had forbidden singing as such in the shower. But the "logic" of his argument actually commits him to a prohibition on singing, speaking, or otherwise uttering biblical Hebrew anywhere in a bathroom (whereas singing, speaking, or otherwise uttering any other language, including modern Hebrew, should be just fine). That's much harder to fit with your interpretation.
I see. Perhaps the Rabbi is establishing a guiding principle of keeping separate certain bodily exposed activities from religious activities that in his religion require cover.
Though it is true that speaking Hebrew in the bathroom wouldn't necessarily be a religious activity, the prohibition makes a point that the follower can relate to all things. The follower won't try to find workarounds but will take the principle to heart.
If this is the case, the Rabbi might think the prohibition is quite subtle.