Yes, Buddha Dharma is the only way to attain Nirvana. Nirvana means No Raga, No Dwesha, No Moha permanently, thus no rebirth, no life forms ever again. You cannot find the way to attain Nirvana in any of the other religions from the entire universe.
No, this isn't boasting. This isn't overestimating. This isn't coming from over-believing or stuff along that line.
It is the truth. Because Tripitaka Sutra contains that the Lord Buddha himself said that Arya Ashtangika Margaya — Noble Eightfold Path — is the one and only path to attain Nirvana.
Also the Lord Buddha isn't on the level of other Religious Leaders. The person with a goal to become a Samma Sambuddha has to perform various actions not just 1 day, or 2 days, or a month, or a year, but countless lifetimes. That's one of the reasons why Buddha Dharma is the only way to Nirvana.
And you're right. Buddhism is not for everybody. It's for the fortunate few who has the potential to reach Nirvana. Out of that fortunate few, not everybody attains even Sotapanna phase before they again lose that "Potential" for a long time.
BUT, that doesn't mean Buddha Dharma does not apply to everybody. For example, Karma applies for everybody. Everybody chasing the happiness in this world will taste a ton of suffering and Dhukka sooner or later.
The one with the Mental Illness in your example, became like that because he did something wrong in the previous life. He may have lost the potential to attain Nirvana in this life. But if you were to do one of the following, you can initiate the chain of sequences that will bring him a life that will have the potential he lost this time.
- If you make them participate in the Donations to Monks, it'll help.
- If you chant Pirith with him, it'll help.
- If you apply other Buddhist Pratikarma, it'll minimize the impact of his illness or even cure it + help attain Nirvana someday.
Just like that, Mental Ill guy can too get the benefits of Buddha Dharma + attain Nirvana if he can spark it with things like above.
Above things also apply to the person with Tinnitus. Buddhist Partikarma can help him minimize/cure it. Also Meditation is not required unless the person has attained Sotapanna phase. This means Tinnitus person still can become Sotapanna if he listens to the right Dharma Sermons.
"Prayer" in other religions cannot make someone attain Nirvana. Pirith & Gatha Chanting in Buddhism can be loosely compared to Prayer.
Buddha Dharma applies to everybody from Humans to Dogs to Dewa to Brahma to Ghosts. So it's not unrighteous. Those who became unfortunate to not even hear a Buddha Dharma Sermon became like that because of trying to chase that happiness by doing wrong things.
Buddha isn't responsible for that. But Buddha explained why that happened to them. This way YOU can avoid it happening to you next. YOU can avoid it happening to others.
And yes again, Buddhism is the sole path to Nirvana not just to stop human suffering, but suffering you experience as all life forms such as Deva, Brahma, Ghosts, Animals etc.
To attain Sotapanna, you only need to hear the right Dharma Sermons from a monk who's already attained one of the 4 phases — Sotapanna, Sakrudagami, Anagami, Arihath. To attain other phases, you can start meditation to reach 4th Dhyana and then brainstorm various Dharma Facts as Karmastana.
I recommend learning Buddha Dharma deeper, and then you'll realize what I said is true.
Hope this helps!