One thing to keep in mind is that we have our quality standards for a reason. Stack Exchange exists as a place to ask questions that are useful for the general public, and get answers which provide solutions to those questions. We prefer generalised questions that will have a use beyond an individual but there are some exceptions.
Additionally, we are not a social network, which is why there are no private messaging features.
Here are some answers to the points you've made:
I have come across many questions on the site that are very trivial and get closed.
Some examples would be useful, we do have some relatively simple questions on the site that result in high votes and high amounts of views, so generally if such a question gets closed it is for another reason (like being very low quality, speculation on new features or unreleased content or developer intent), or being a question that is so straightforward that finding an answer to the question is literally a case of typing your question into Google and clicking the first link.
However, it is still helpful for the person asking the question to get an answer.
I agree, and we try to answer any question that meets our quality standards for this exact reason.
I was thinking there should be a way to continue helping a person with a trivial question privately.
This functionality already exists, it is called chat and it's available at chat.stackexchange.com - the only requirements are that one of the people involved in the discussion has at least 100 reputation (to create a chat room) and that the other users have at least 20 reputation (to be able to chat). Since all users have at least 1 reputation, this is a case of generating 19 reputation, which is relatively easy. If you're struggling to generate the 19 reputation required to chat, try improving the quality of some other questions and answers by using the "suggest edit" feature.
I don't understand the insane strictness of this site. I guess the site cares more about itself than helping people.
This has previously been covered in a Stack Exchange podcast; essentially while you may think that allowing your one trivial question to exist is for the betterment of the site, what would happen over time is more and more people would use your one trivial question as justification for why their one trivial question should remain, and before you know it, the entire site would consist of trivial questions, both hiding and discouraging the actual in depth questions from existence. Over time, this would essentially result in the failure of the site.