Bear in mind I started my singing long before the ability to tape yourself was as simple as holding your phone in front of your face, so this might feel a little...ancient ;)
You have to develop a sense of what you sound like when recorded... without listening as you record. That's the only way to gauge 'how you sing'.
Even with headphones, you cannot accurately judge what it will sound like to others, you have far too much of your own head resonance going on at the same time.
I think if you try to train this way you will over-think it & struggle, long-term.
Learn to sing, then learn how to record it. They are different disciplines.
Your voice on playback is partly vocal ability, but partly mic technique & recording technique. Sticking a phone or even a cheap mic in front of your face & singing into it will be a disappointing playback experience [and not because the singing was poor].
Let me go a bit old school & give you a reeeeally quick précis of 'how I learned to sing' ... see if it makes sense
[late] 60's - school choir.
70's - amateur pop/rock bands, musical theatre.
80's - damn... got a record deal - had to sing in the studio with headphones on for the first time. First time I'd ever heard my own voice on playback, after 15 years' practice without hearing it. It sounded like someone else, not me.
I spent the next 5 years learning to tune/trim what I heard in my head to make what came out on tape 'what I wanted to hear'.
End result -- a successful career as a session singer, doing whatever style or nuance I was asked.
25 years later, I barely do more than a bit of 'fun karaoke' on holiday... but I can still call up whatever type of vocal performance the track needs - because I know what the voice in my head has to sound like for it to be 'right' to the audience.