Let's say it all together now, Rin wants to beat his brother to prove himself, not out of hatred.
I feel like this sentiment got kinda lost as the fandom got larger/the anime was released. Which I understand because there is a lot of emotions that are going into this dynamic, but let's look at what we know.
Rin wants to go back to the past, where he was his brother's number one person, his confidant, partner, friend, secret keeper, duo; this list goes on because they were siblings that did everything together. Rin, truthfully, is projecting his desires of returning to the past onto soccer. (Soccer for Rin is very symbolic, in contrast to Isagi who just has a clear love for the game.) He wants to go back to the time where he was Sae's exception to everything. Sae was good at soccer and didn't think anyone was worth being his partner, but then Rin became the exception. Sae could be harsh, yet he'd show gentleness towards Rin. Sae thought others were childish and silly, yet when Rin acted like a child he would care for and comfort him.
We can see that the Itoshi brothers had a bit of a codependent relationship, the term may be strong but you get the point. While they had individual dreams, skills, and experiences, once Rin also started playing soccer everything in their life became enmeshed with each other. They continue to develope and grow their bond until they are forced apart, but they're obviously still constantly thinking of the other. So, Sae pursued soccer wholeheartedly living not only his dream but his brother's as well but *thunder clap* Spain happened. We don't know what happened but it made him realize that he could not achieve his dream, much less two, or he realized where his true talents lie, either way heartbreak was awaiting Rin.
The first time Rin was not special in Sae's eyes was when Sae challenged him in one-on-one. It is exactly what we see, a test to prove himself, no stops or aid from anyone or Sae. This was a test for Rin's independence. (Notably, one that required him to face the least amount of embarressment when he fails.)
We often see Sae carrying the belief that no one in Japan is capable of becoming a worthy striker, but the subtext is 'not even Rin'. This is the nail in the coffin for their relationship, the initial wedge being Sae leaving for Spain and the distance between them. Keyword, their relationship, so their bond as brothers not their ability to be partners in soccer. BUT soccer becomes the proxy for said relationship. Hence, why Rin believes that he needs to beat his brother for their relationship to go back to how it was before because then he would be Sae's exception again. It's not Rin's goal to lead Japanese soccer into a new era or to become the best player in the world by other people's standards. He just can't stand being ignored and pushed to the side by his brother, and to him, it was soccer and his soccer abilities that caused the change in their relationship.
Another thing is that Rin would not view the soccer as a battle field if all he wanted to do was beat his brother out of sibling rivalry. This is serious to him, its not just a game, or just a competition. His life purpose is to prove Sae wrong and show that he is still an exception; Rin can be the striker to change soccer in Japan for Sae. (Or at least impossible to ignore but we all know that the Itoshi brothers carry an 'all or nothing' attitude.)
And then Sae acknowledges Isagi first....yikes
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