See my answer here. It's an in depth answer about having a rational approach in Islam.
About your assumptions:
truly reading/following the word of Muhammad
No Muslim should assume such. There are a gazzilion disputes among Muslim scholars about many verses, many aspects of Prophet Muhammad's life or its application to today's world.
they don't criticize neither the source of the information or the
information in itself
We do criticize, we should be critical of what comes to us. Read my linked answer.
why should we not tolerate each other exactly as we are
We shouldn't force them:
There shall be no compulsion in [acceptance of] the religion. The right course has become clear from the wrong. https://quran.com/2/256
Let them have their religion—if they don't want to listen (don't force them, let them enjoy their own ignorance)
For you is your religion, and for me is my religion.https://quran.com/109/6
That shouldn't be the case
providing me with the logic behind your reasoning or simply denouncing
logic in favor of religion.
A group of Muslims just follow leaders (good or bad, unintellectual or intellectual, pious or evil). I'm sure a group of atheist, Christians, Jews, Hindus just follow people. People are people. Muslims aren't from Mars or anywhere.
The other group that are intellectual, don't just follow they are critical
So basically to answer this part:
The group who just follow, may just have leaders who evil, have an agenda or are simply very unintellectual, which I highly doubt. I mean what kind of a person would prefer violence?!
The other group who are intellectual denounce violence completely.
But why violence can be interpreted by mistake?
Because the prophet himself was attacked on numerous occasions. There were many assassination attempts against him. Why? Well imagine there was tyrant who has been killing people for years. Have control over all the economics. Suddenly a humble, loving leader rises and people find hope in following him. Well what does the tyrant do at this moment? He's going to bribe him at first. Then he's going to threaten. Then's he's going to attempt to assassinate him. If none of that works then he will go at war against him. If the tyrant can't take this loving person of the surface of earth, then he's doomed. All the people will go away from him and he will lose his position and wealth.
Because of this turbulent time, there were harsh verses against these people. Some of them even wrote peace treaties with Prophet Muhammad then broke those treaties by killing some Muslims. Then a verse came and said, kill them where you find them.
Imagine if America was having a war with N.Korea and then they wrote a peace treaty. Then a few days later N.Korea soldiers kill some Americans in S.Korea. They are basically peace breakers and never to be trusted again. But just as an example see what America did against the middle east after 9/11. They literally put the entire region in chaos.
Another reason was that in the early centuries after the death of the prophet, many Muslims were merely just followers of the prophet. They were only following Islam for their own benefit. Islam was in power and many people joined it. It's like a new revolution where there's a power shift. These followers brought fear to the Muslims.
So 1. applying the verses of the prophet's situation to this time and era is a mistake 2. applying the wrongdoings of other Muslims is another mistake.
But really that's not the problem. The problem stems from their lack of collective understanding of the life of prophet.
Did the prophet participate in wars? Yes in a defending manner. But more than anything he's known as a prophet of mercy. He pardoned and freed many of those who fought against him before in the Conquest of Medina. He took control of a city without any bloodshed from battle.
The forget that the prophet said: “I was sent to perfect honourable morals.
انی بعثت لاتمم مکارم الاخلاق
and instead think that he has just came to bring everyone to Islam by force.