“I never realised how hard actual mathematical proof would be. I’m currently studying different types of proofs in my calculus topic (which is insanely hard). I see what you mean by saying A-level maths does not teach you actual proofs, because the proofs they provide at uni are completely on another level than A-level”.
Don’t let that put you off studying maths at uni! It’s good that uni maths courses teach you to question assumptions and arguments deeply. All students find it a culture-shock at first, but they manage. After a while it’s exhilarating, learning a new way of thinking.
Part of the reason why I give over a week of Year 12 Further Maths to studying proof, though there’s nothing about that in the Edexcel textbook or in the exam spec, is to prepare students a bit for the culture shock.