Thanks for your question. I hope my answer brings you confidence that you can help yourself and a direction in which to begin.
To every situation, there are always two sides - and on this occasion I am wondering if you have been able to explore the positive aspects of being an only child, to balance out your overwhelming evidence that you are missing out on something? Have you spoken to your parents about supporting you with this? I understand it might be difficult because they are part of your problem, but I think you will need support from somewhere to help you.
Having said all of that, the strength of your feelings resulting from this appear to be quite strong (disdain, revenge, jealousy, desperate) and you mention that you think this may have been building for a while? I would like to explain a model that I use at work for you so you can see how this may have developed for you - and then tell you what you can do to help yourself with this. This is one method for you to consider.
Sometimes, particularly when we are small, we can pick up on information from the world around us and it forms into strong beliefs in our subconscious mind, where it is often unnoticed, but its power can be very strong. Many of these beliefs are what protect us in the world and enable us to develop (for example, 'I am loved', 'I have a purpose'). Many of these beliefs shape our life and we never notice them. However, sometimes a belief can be strong and doesn't fit with our world as we grow older and it begins to create problems - for example, someone who has developed a belief that 'I am not safe on an aeroplane' as a child may find that they cannot go on holidays abroad as an adult for fear of flying - it has become a phobia and therapy can help them to lose the fear. The effects of the underlying beliefs are sometimes not noticed for many years, but eventually can be recognised by strong feelings and behaviours being triggered by current events/people. I hope this makes sense to you?
There is another statement that you mention that suggests that this may be what has happened here - you say 'I know it sounds silly but' - and this can often be an indication that your adult self is making a grown up opinion of the childish beliefs within you - your adult self can recognise that this belief is silly, but its affects are still strong and you cannot find any answers to help you out.
Now, what can you do about this? If you can afford it, or your parents are willing to support you financially in working this through, what I suggest is that you find yourself an EMDR therapist who is trained specifically to work with young adults and work with that person in order to recognise and let go of your strong beliefs in relation to this situation. The therapy taps into the memories that are subconsciously held in your mind and allows them to be processed away, in the same way that our anxiety from daily life is usually processed away during Rapid Eye Movement (REM) sleep at night. It is believed that incidents of high emotion (either positive or negative) become lodged in our subconscious memory and we can tap into them "as if they were yesterday" - feeling the emotions and sensing other parts of the memory, such as textures, sounds, smells, etc. When the memories are positive (good exam results, winning a race perhaps) we can recall them and use them to empower us, but when they are negative, we push them down and supress them so they don't hurt us. The more they are pushed down, the more they 'shout' so that we hear them, and the more we notice the 'symptoms'. You may be able to recall incidents that may be relevant from your past, or not, and the therapy can help you to recognise them if you can't. They don't need to be huge - they can be simple incidents that you interpreted as a child, from a child's perspective, at the time. The therapist would help you with every step of the therapy and you would be able to let some of your beliefs go.
This would obviously not produce the sibling that you would love, but with a changed perpsective, you would be in a better place to accept this and to consider the value of other alternatives and perhaps the positives of being an only child.
If you search for 'EMDR therapist adolescent specialist', you should find one local to you on the internet.
I do hope this answer is helpful for you. It will appear twice, which is something I am trying to fix. I think there is a problem with my pay button at the moment, but if this answer is good for you, please try pressing the 'accept' button and see what happens. If it doesn't work, I would appreciate you contacting JustAnswer by email at email@example.com and letting them know.
Best Wishes, I hope it goes well, Sarah