Don't feel so self-conscious about it. What if your parents were also quiet and introverted and that's the way they were talking with you? Or/and they wouldn't talk that much to you at all so your emotional and stimulation needs were neglected since the time you were born. People who don't talk quietly like us grew up in a different environment. If their environment was the same, they would talk just like you. Don't compare yourself to them because that's not fair at all. This is what I'm like and this is why I had and still have same problems you talk about and, of course, that's why I have social anxiety because even social anxiety on its own is just a symptom of such or other problems in childhood in many cases. So this is not a ''voice problem'' per se.
I don't know if speech training can do much in the long run because you'll start feeling insecure again for different reasons and your habit of talking quietly will go back and you wouldn't be able to control it because of emotions. Maybe not, maybe your voice technique really helps to be in control 100%, I don't know. I don't even know if that's possible without certain emotional component, no matter how good the technique they offer is. But I know that's what would happen in my case. I know that training like that would only help me and would worth it if I got generally better psychologically. I got a little better so far due to therapy and someone else who's being often emotionally supportive to me, but it's not better enough. Btw after I had a really good session (which happens really rarily to me, I experienced just a couple of such sessions during the whole time in therapy), I noticed how I was automatically breathing deeper. I don't remember breathing like that so it was a little unusual. I had to be in public after that session and was having a panic attack, but somehow I was able to calm down my breathing by remembering the support and involvment of the therapist on that session. It didn't last for a long time. Only for the whole evening. So yeah :/
When it comes to me, I'm not only super quiet, but I also sound hoarse quite often. It's because my whole body is tense and this is just another habit just like the quiet voice one. It's (often unexpressed, unconscious) emotions, feelings manifesting themselves in the body. I was never aware of it because for me it's a ''natural state'', but other people have been saying that to me and pointing that out since childhood. When they would tell me to ''relax'', to ''put your shoulders down'' etc, I would not know how to do that because I don't know how it even feels. It's so sad people judge you by themselves and they think other people are or should be just like they are. It's a simple ignorance you should brush off. But then when you're not aware of why you're this way you believe them and start feeling worse because you feel abnormal and helpless. This is how I felt before I found out about the reasons just a couple of years ago.