I can totally relate. I never really learned singing or hearing pitch differences as a kid and although I’ve been trying to relearn this for years now it’s still hard and by far not 100% correct all the time but much better than it was 5 years ago. For me trying to learn to sing on pitch helped a lot, but just like any other skill it needs lots of practice and dedication (in addition to learning the language itself). For learning Japanese itself you have the advantage you don’t have to hit a specific note or something, just the direction is important and even if it may not seem like this to you in the beginning, you can get pretty good at this quite fast.
When speaking, the main challenge is to decouple the pitch change with the volume and length change I know from all other languages I can speak. And that is where I also see a problem that can happen when listening to music. To take the “Can you feel the love tonight example”: Yeah, the “can” is higher than the “you” but it’s also longer and louder and so it just feels stressed (at least it was like this for me) and it’s hard to discern which part of the stressing is the pitch. If you want to get used to the Japanese language, I guess it would be better to listen to Japanese songs or watch Japanese shows to get used to how the language sounds.