I've been thinking about this question since it came up. My weakness, as you know, is at the opposite end, getting a feel for the underlying drumbeat so to say. Singing, I think (?), is supposed to be for phrasing.
The only and then main thing that I had for a long time was singing. When I learned to play the piano, I sang the melody from the page in some rudimentary solfege, and then played what I heard from the page. The same thing went for recorder, guitar, or any instrument I could get my hands on. There were no lessons, and there were no models. Yes, I had a strong feel for phrasing. It was also undisciplined, with a rubato untamed by underlying pulse, and there were certain "cliches" to my music, because none of it was controlled or conscious. Did it give me something? Probably. But it also is not the bee's knees as it seems to be made out to be.
If the question is phrasing, can't that be done in spoken proclamation, as in a poem or an orator's speech?
A weakness in singing being the first association, as it was for me, is that the piano is not voice propelled by breath; it is percussive. If you hear a single note swelling for three beats, the piano doesn't do that. If you put energy into that note for three beats, with piano you have to let go of that note in a single percussive strike. If you're a singer by nature, you have to remove your voice from the equation so to say.
What about the fact of chords? How do you sing or even hear chord qualities? Answer: you don't.
I also learned over time to create some things artificially and cerebrally. Gradually slow down this passage: extend that note a little bit more (agogic accent) - If your melody is quiet, create the crescendo diminuendo artificially in the left hand giving the illusion of it happening in the right. If singing had a role, it was only a partial role.
And then, nobody has the range of a piano. Might speaking in phrases or chanting beats scat-style also play a role?