I’ve never written a college essay so these are just suggestions, but to me, though well written, your essay comes off a little too passive. There are some things that I would rephrase, but overall it’s pretty solid. I put all my suggestions together at the end (plus a few minor additions / rephrasing) , so most of this is just explaining my thought process so you can make any modifications you need :D
First off, there's technically nothing wrong with “But in my opinion”, but here I would change it to something along the lines of “But I have always derived more joy from helping others accomplish.” or “And I have always felt that the most satisfying form of accomplishment is found in helping others achieve.” These also help make it a bit more personal.
Since I just used the word ‘always’, I would change the next sentence to “I can’t recall a time where teaching was not prominent in my life, and it has always been my favorite thing to do.”
This might just be me, but “My favorite game has always been school” did confuse me for a second when I read it. A phrasing such as “My childhood favorite game was school, and there was nothing more thrilling than getting new teaching supplies from the Dollar Store.” could help clarify. I also think the essay flows better with the two sentences combined.
“But I really wanted a real one” feels passive and slightly informal, I would take out the ‘really’ as it acts as filler and keep it as “But I wanted a real one.”
Similarly with “I needed someone like a younger sibling”, I would take out “someone like” and leave it as “I needed a younger sibling”, or alternatively rephrase to “My older siblings never agreed to play, and I was left wishing for a younger sibling.”
Same with the ‘just’ in “Just everything by myself”, very filler-y, you can cut that out.
“I finally got one and I hated it.” made me giggle a bit because I couldn’t tell if you were talking about a doll or a younger sibling with the ‘it’. I’d change that to “I finally got one, and I hated her.”
The next part of the essay is fine, but for some reason I truly cannot identify, the general lack of contractions became super obvious and interrupted the flow. I’d drop a ‘wasn’t’ or ‘doesn’t’ or two in there
And finally, the phrasing in the sentence “She taught me to be patient because not everyone can learn something right away.” is a little off, I would change it to something like “She taught me patience, and through her I learned how best to help when someone does not immediately grasp a given concept.”
The feeling of accomplishment is arguably one of the most incredible, and I have always felt that the most satisfying form of accomplishment is found in helping others achieve. I can’t recall a time where teaching was not prominent in my life, and it has always been my favorite thing to do. My childhood favorite game was school, and there was nothing more thrilling than getting new teaching supplies from the Dollar Store. I loved my stuffed animal students, but I wanted a real one. My older siblings never agreed to play and I was left wishing for a younger sibling. I was sick of playing dolls by myself, playing school by myself, doing everything by myself. When I was 10 years old I finally got one, and I hated her. She couldn't do anything. She could not play dolls or school or hide and seek, or even talk. I had to learn patience and teach her these things. By the time she could do the things I wanted to do when I was younger, I was no longer interested. She whines and stomps her feet when I do not want to play. She doesn’t understand that I do not like playing anymore. Despite my complaints about my sister and how she annoys me, she really shaped me as a person. She taught me patience, and through her I learned how best to help when someone does not immediately grasp a given concept. She taught me to be stern in certain situations to help make the right decisions. She taught me to be a leader and what it takes to be a role model. She taught me who I am going to be.
Hope this was helpful, and good luck!! <3