I'm so sorry for your loss.
The thing is, we have a system where the midwives won't be personally known to your wife. She won't know what their views are, how compassionate they are, and won't have any familiarity at all, in a very vulnerable moment.
I needed my husband there in labour. But what I needed was for him to sit down, shut up, and let me focus on birth, and that's not uncommon when things go well. Your ex's aunts had no right or place to make you feel like you were redundant; it was your baby, not theirs, and you were closer to the mother than they were - their actions probably made things worse for the two people most central to the situation, and that was awful of them. But a doula is a professional, albeit less skilled/trained than a midwife, and part of her professionalism will be knowing your role is more significant than hers, and that you are an important element to the birth for both mother and baby. It's not an overbearing relative seeing "their baby" in the form of the mother suffering, and treating adult emotional ties as less significant - it's a skilled, knowledgable birth attendant who will be better informed than you or your wife, and not in the throes of labour, so she can advocate for what your wife seems to want/need, and actually also explain to you what's happening because she won't be physically intervening with the midwives' actions. A good doula is a woman-centred supplement to the medical team, not a replacement for a birth partner, let alone the father.
I actually had my mum there at my first birth as well as my husband, and it was just as well. My midwife was herself very pregnant and it was the middle of the (weekend) night, and so she was monitoring me via thermometer in the birthing pool without noticing that an oral temperature might be skewed a little if the patient was sucking ice cubes between contractions. There was a change of shift, and the new midwife panicked because the water in that pool was in the mid-40s. I'd been relying on very hot water and gas and air for pain relief, and nobody had noticed how hot it had got. A baby born into that would be in a lot of trouble - 36/37 is the recommended top temperature and most adults would find mid-40s scalding hot. She tried to get me to get out, but I was in transition, off my head on gas and air and one continuous contraction by then. So she got more aggressive and shouty from panic, at which point my husband started shouting at her to back off me and to stop talking to me that way. Meanwhile, my mother quietly turned the cold tap on full blast and started swishing the water around. They both turned, stared at her, and the midwife went "oh" and then rushed over to start filling the water jug with cold from the sink and adding to it. Water cooled, easy delivery, baby thriving and Apgar results all good. And I still needed my husband there, and the photos of his delight holding our PFB and his excitement at the moment of birth itself "he's here, he's really here!" will always remain with me. I don't really remember my mother's presence because she just wasn't on my emotional radar - my husband and baby were. But her being there was undoubtedly hugely helpful.
I think my point is that having a baby is such a huge, personal and shared thing between the parents that nobody else really matters, as long as the birth attendants are doing their jobs right so things progress safely and smoothly. My mother is fairly emotionally detached in moments of crisis, after a boarding school from 7 childhood, which is hard in many ways as her child but quite brilliant at moments of high stress: she was the only person in that room to think calmly and sensibly because it wasn't her professional neck on the line, and it wasn't her baby. Yet her presence wasn't remotely as important to me emotionally as my husband's. It was our baby being born, and he is the person I love most in the world along with those babies. Nobody can replace or undermine or threaten that role because he created them with me. And oddly, I think if I'd had a closer bond with my mother she'd have been less use to me in that scenario. She has a medical background and fabulous professional interpersonal skills, so I knew she'd be a wonderful advocate because she'd go into that professional mindset. My husband would love me, worry about me and be there for me in a way nobody else ever could be - and actually that's reflective of our relationship generally, not just in the labour. But funnily enough, my husband is a very calm, gentle man and I'd wanted her there as she can be tank-like in her assertiveness, but when I was threatened my husband blew up and she was meekly circumventing the argument to fix the issue practically. Their roles were different - one was practical, the other emotionally supportive, from just from sitting there and quietly reading a book to clearly and angrily stating to the midwife that I was a human being in immense pain who deserved basic courtesy, whatever the situation. I needed him there emotionally, not her. But when there was a minor crisis, as it turned out I needed her there too in that capacity.
I do think your feelings are completely natural, after experiencing such a horrendous bereavement. It would be odd if you didn't have such feelings, actually. But the great thing about a doula is that they're a familiar, yet detached figure. An informed and experienced figure. She'll never be your wife's spouse or your baby's father, and after the birth period ends you'll probably never see her again, any more than you will the other birth professionals. But meanwhile, she can support both of you in ensuring the sort of continuous, one-to-one care we all want and need, but the NHS just can't afford.
It's also worth talking with the doula and your wife about not wanting to be pushed out, as you undoubtedly and unfairly were last time. You could even have it in the birth plan that you are not to be requested to leave under any circumstances. Try to be open about your fears and anxieties, because while your wife is the one to labour and birth, you've lived through something so terrible associated with that process, and you would be a robot or a sociopath were you unaffected.
I do hope, and actually believe, that witnessing a normal delivery will be pretty healing for you. What happened to your baby last time was such a tragedy, and in this country a very, very rare one. Most births are simple, straightforward and uncomplicated, and even the complicated are almost always safe. Your loss was so very, desperately unlucky, and hopefully witnessing a good birth with a healthy mother and baby will heal some of those hurts for you.
I wish all three of you the very, very best.