This is not how people get HIV. They mention theoretical risk but that is not actual risk. NO ONE has gotten HIV this way in a single, documented case in all the time of HIV and studying it. Experts in the field of HIV guide the answers we give on this forum and these are the only risks that lead to HIV transmission. They include having unprotected vaginal or anal sex with penetration or sharing IV drug needles. Saliva and air inactivate the virus. So, deep kissing even with sores in the mouth would not result in your getting HIV. No, you are not seroconverting. Symptoms do not work that way and are NEVER used to diagnose HIV.
We are in the middle of a pandemic as well as other viruses being prevalent. Sore throats are rampant with viruses. If your sore throat persists, get a covid 19 test. Wear a mask and keep your distance from others.
Your situation involves personal contact with an object in air ( braces, mouth, lips, sores, etc.). No hiv worries, because you can't get hiv from personal contact except unprotected penetrating vaginal or anal, neither of which you did and you didn't share hollow needles to inject with which is the only other way to acquire hiv. Analysis of large numbers of infected people over the 40 years of hiv history has proven that people don't get hiv in the way you are worried is a risk.
HIV is a fragile virus in air or saliva and is effectively instantly dead in either air or saliva so the worst that could happen is dead virus rubbed you, and obviously anything which is dead cannot live again so you are good. Blood and cuts would not be relevant in your situation since the hiv has become effectively dead, so you don't have to worry about them to be sure that you are safe.
There is no reason for a person to test when they are safe. The advice took into consideration that the other person might be positive, so move on and enjoy life instead of thinking about this non-event. hiv prevention is straightforward since there are only 3 ways you can become infected, so next time you wonder if you had a risk, ask yourself this question. Did I do any of the 3? Then after you say "No, I didn't" then it's time to move on back to your happy life.