I think the "message [you] are getting" is coming from your internal world and not your therapist. Shame is a sticky widget and the traditional understanding of it is we feel shame for "who we are" as opposed to "what we've done" (guilt). Thinking about that distinction has been helpful for me.

I think if you open up your willingness to consider thinking about your therapist's refusal to reply to your emails, which includes accepting what he says about it at face value and which, IIRC correctly, you agree has actually happened -- you have been able to access "young" emotions more readily in therapy. So your therapist has actually said the reason for his choosing not to reply to your emails is to facilitate something important (emotions) happening in session. He has not said you are "bad" for wanting an email response and he has not said that the perhaps underlying emotion, which comes from a "young" place, is bad. Those are your interpretations and probably come partly from some kind of distorted thinking about what it means to have needs (as opposed to being so self sufficient and independent, which you've said about 10,000 times is not who you are).

There's a ton of stuff that seems possibly related to this issue for you, but I don't think you can experience it if you hold on so tightly to your preferred way of thinking about yourself and what this issue is actually about. Open your mind-- and at least it seems to me that you have a fine one infinitely capable of alternative ways of thinking-- ask yourself questions rather than narrowly defining this experience.

I do resonate with the experience of shame related to the label of "young," and I can precisely recall a moment where my T characterized the way I was talking about a grown-up experience of pain. It was more than 5 years ago.
"That comes from a young place of understanding," he said. And I had said this exact thing to my actual doctor. My face flushed with such warmth I thought I was beet red, which through my dark olive skin tone, takes some effort. Something really familiar ("I am bad") stuck with me for some time. For me, as anyone reading this might have guessed, the shame was attached to thinking I was bad for being physically hurt by abuse, which is super common for CSA survivors. I had tapped into my childhood experience with pain and being hurt meant I deserved it, that it was punishment for something I did wrong. I thought I'd already resolved my self-blame for what had been done to me. It is often true for me that childhood stuff is triggered by adult events and experiences in unexpected ways.

So if I try to put myself in your shoes (which I am sure I will do very imperfectly, but I hope something might be useful anyway), I would ask myself, "Why feel shame when I ask for help and it is refused?" This is how I would characterize your email, that you are asking for something that will help you-- as opposed to a reply per se. Any current situations in your adult life where you've asked for help outside therapy and you've been refused? Or have you constructed your life in such a way (something I identify with) so that you just don't simply ask for help? When I was married, I pretty much did everything myself, which greatly benefitted my spouse, rather than ask him for "help" with the baby or the house or the scheduling or even to make the time to talk to me about something that might be difficult. He was great at talking about anything that didn't personally affect him but somehow was super busy when it was about him When he became ill and people wanted to help, so many people said something like "Anne, you are a hard person to help. Let me do something for you." I had to learn how to let people help me. And sometimes I'd ask, and they would say they couldn't, and I'd be all like but I hardly ask for anything and then I do and look what happens. So I had to learn about asking and being okay with getting the actual answer rather than the answer I want.

So I think I get at least something in the neighborhood about how ****** this feels, and I suspect if you take a look at how you relate to other people, you might find some present and historical answers about what it means to ask for help. I think a child who reaches out for help and doesn't get it experiences pretty deep shame. Whether it is refused directly or whether the request just isn't understood (the later was true for me in the sense that I asked for help but being a child who experienced trauma (which messes up one's ability to articulate what has happened and what kind of help you need) I couldn't get other people to understand me. I was vague and couldn't say directly what I needed. So when help didn't materialize I took that as some kind of message that I didn't deserve help, and if I didn't deserve it, it was because I was bad.

Maybe this doesn't make any sense to anyone but me, but there it is. I think you're stuck because you are having trouble seeing this in a more global or historical or other way. The thing is not the thing. It's not about your T's refusal to reply to your email and whether he's right or wrong. The thing is about what this means to you, what you feel about it, and where that comes from. And this thing is big, complex, and has the potential to resolve itself in ways that create deep change in you. But only if you deal with the actual thing.