ok. I think that the very first and most important thing is to stop finding your voice cringeworthy.
Learning to sing is about noticing, hearing and bringing out what is good, expressive and interesting in your voice. You do that by training.
Anybody who tells you that "you don't have a singing voice" when you are starting out, most probably lacks imagination. You mustn't become one of those people.
I cannot hear anything in your sample to suggest that you "don't have a singing voice". The vast majority of us develop "singing voices" through training.
Go for it!