If this isn't possible in your case, then you should try and prioritize emails thanks to indexing, virtual folders, and Gmail's Priority Inbox. Don't disregard the power of that new feature from Gmail. If used well, you can attain inbox zero for your priority inbox, which is better than nothing. All of the notifications and emails that you still want to keep in your inbox are still there instead of being archived.
If you can't use Gmail's Priority Inbox, then you can star, index, and label your emails manually. You can also create rules so that your inbox does this automatically. Then, you can try and ignore some of these emails while addressing more important ones. In a sense, you're creating your own priority inbox. We've found that it's just easier to use Gmail's service than to try and do so by yourself, but if you've got no choice, then you can try this.
There is no need to contact every single person that sends you an email, which is why you need to prioritize. In fact, I do something similar, but I just press archive or delete when I'm not interested in replying to some emails; not every single query needs an answer. Naturally, I do so for low-priority emails, not important stuff that absolutely needs to be addressed. It's all about finding some sort of balance.