Competing sources of media help with that too. They're not all uniformly sensationalist.
It's hard to be a relevant voice in news media and not be sensationalist. Even so-called "quality newspapers" are not free of sensationalism, given how certain political conflicts and disasters get more attention then other events with significantly higher human costs, or similar events closer to home.
It makes you uninformed, perhaps, but it doesn't endanger political rights unless taken to an extreme (eg one media outlet utterly dominating over all others). Again, competing media, competing ideologies.
The same fallacy is made when people argue for capitalism and free markets: Competition saves us all! Yet, this is not what make free markets good: What make free markets good is that people have the option of making the right choice, despite it also means they can choose the wrong option and people frequently do so. The same applies to freedom of expression.
You're right, political movements can't grow without the news media. Precisely why the free press is such a danger to authoritarians. Media is all about organisation; organised distribution of viewpoints, and the organisation (consciously or unconsciously) of people subscribing to those viewpoints. Not necessarily a bad thing; could be, but not necessarily.
Political movements can grow through word-of-mouth when immediately necessary, as in before the French revolution. After the revolution is concluded in either way, the movements disperse, as they should. Except when we have news media, in which case anachronistic political grouping will be perpetuated by the media. People are now divided into crisp political groupings that otherwise wouldn't exist and these are made in the pursuit of profits and political gain.
Also note that press freedom is somehow very limitedly correlated to democracy and overall liberalism: Italy and Israel (definitely my favorite country to talk about) for instance have some of the worst press freedom rankings in the 1st world, yet have roughly the same political liberties as several of the best ranking countries in press freedom. Likewise, Turkey has jailed far more journalists than North Korea, but I don't think anyone with a sane mind would say that North Korea is a more free place than Turkey.
Basing political views on personal experiences can be just as dangerous. You talked about the importance of looking at the big picture earlier, but most of us can't form a big picture view of the world from personal experiences alone simply because we have a limited range of personal experiences.
We need networks and media to inform us of experiences of others outside our own daily lives. In this organised media is in fact a step up from vague rumours since at least some of the stories are fact-checked.
Basing political views on personal experiences can only go awry when we are seduced by the same thing that make news media inherently bad: Focus insensitivity. Hell, even a thread about it has been opened on the OT! Just one negative story, experience etc. tends to dull our view of reality, despite all the past events that we are unaware of passing.
Yet, personal experiences alone give a better vision of the big picture than the news, which is superficial and usually comes with topics such as "Turkish guy shoots German guy" which can easily lead innocent but unwitty victims into believing that Turkish people have an innate hatred for Germans. If you witnessed the actual shooting, you'd probably be hardly interested in the ethnicities of either the perpetrator or the victim.
The other ingredient in understanding the big picture will be intense immersion on the subject, by carefully reading and interpreting the historical context of the subject, like you certainly have. Only dedicated sources on history will be able to give the knowledge required and not newspapers.
Then wouldn't that problem be worse without a free press?
Certainly, but I think the point is also that it is a general mentality problem. If people were to take actions and storylines depicted in entertainment media like video games or movies at face value under every circumstance, a lot of people would be inspired to go on shooting sprees. That only a few people do this has to do with the fact people are properly trained in how to properly process entertainment, ethically speaking.
Strangely, there are relatively few people who want to ban newspapers compared to people who want to ban entertainment media, even though I consider it much more reasonable to call for a ban on newspapers than on entertainment media. Like I said, freedom of expression is much more important than freedom of the press, which is only a very small subset of the latter. Freedom of the press is to freedom of expression what freedom of offering tarot services is to free enterprise.