JB_PT I wonder if getting comfortable with physically singing is a helpful (or necessary) first step.
Do you have any experience singing in large groups (school choir, church?) or in the shower solo, or have you grown up not really ever singing?
I apologize if that sounds judgmental; I don't intend it that way. The thing is, I've only recently realized that more and more of the younger generations (I'm nearly 50) have had much less in the way of singing/vocal/choir experience than was standard when/where I grew up. In a middle-of-the-road Midwestern suburban school district, we were expected to sing as a class multiple times a week from first grade onwards, as a matter of course. So it that sense, it didn't matter how bad someone's voice was; they got used to singing in a group regardless. Very few people were ever assigned solos, because that would have been pointless torture for everyone involved, but the expectations for group singing were that it happened frequently, and that everyone was going to do their best to sing in harmony. I think the vast majority of kids also sang as part of a congregation at church at least once a week, so it was enforced on the weekends, as well.
I guess I'm trying to understand if you're not used to singing out loud at all, ever, or if you just don't like doing it solo even if you're alone, or if you just particularly don't like/aren't used to doing it when you play the piano. The prescription for how to move forward is different, in each case.
An example: my therapist has been trying to learn to 'hear' how the melody in a jazz piece should go if he just has the left-hand chords of a lead sheet. I have no understanding of how this works (I was like, what, you're supposed to figure out the melody on your own with no music???), but apparently it's a multi-year project. His teacher thought it would take him 2-3 years to do it, and in year 4 it seems to be finally coming together. However, his "phrasing is too long", which I interpret to mean that he doesn't seem to have a sense for when the music needs to take a deep breath - when his hands should lift. His teacher advised him to buy a melodica - which is sort of like a harmonium or accordion, but the air is pumped by the mouth - to develop a sense of when his phrases should end. Essentially, he plays the melody on the melodica, and when he starts running out of breath, then he knows that the phrase is getting too long. Anyhow, however brilliant this technique is, it generally only works for a student who has some solid experience with breath control while singing, or else the phrases will be too short. People who haven't sung out loud in a long time tend to have a reduced ability to control their airflow to be able to complete an entire phrase in a single breath.
I'm starting to think that the advice to 'sing the melody' may not be that helpful to people who didn't learn to sing at all in school or at home, since humming along to a song with headphones on is a very different activity.
TL;DR: silent or mental singing may not be helpful in playing the piano more musically if someone doesn't already have experience with singing out loud so that they can hear themselves (i.e., no headphones or ear buds). Singing with a bunch of other people counts, but only if you're still able to hear yourself somewhat.
I do want to assure you that the quality of your voice isn't an insurmountable impediment, though. I'm debating sending you an audio clip of me beginning to learn a simple piece yesterday. I listened to some of the clips, and my voice is amazing...ly bad. I de-tune in the middle of notes, I think I squawk in places, I grunt....It's cringe-inducing. In some ways it's not even amusical; it can sound like it's anti-musical. And yet, it works for me - my terrible-sounding voice is a good and helpful intermediary between my brain and fingers, both increasing the push-pull connection between the two, as well as triggering my brain to think about how the music might sound better if I do x, y, or z.