See discussion in https://github.com/piwik/piwik/pull/6541/files#diff-842c76803d19071217437497af04afd5R615

Problem are exceptions like HtmlMessageException in classes like Filechecks https://github.com/piwik/piwik/pull/6541/files#diff-ec1435896b71140d801a0b5d8d2ca125R107 and Session https://github.com/piwik/piwik/pull/6541/files#diff-d1904580a82516edb7ede7a2d4ca3a37R135 but kinda also always using the \Exception.

Those classes do multiple things. They handle for instance file checks or sessions and they "render" / "output" things. This prevents the usage of those classes in different contexts (for instance in Console or in API's you do not want to output HTML) and it makes it impossible to treat exceptions in some cases. Also if you call a method you don't know which error actually happened as all of them throw the same exception. Sometimes you maybe want to catch one type of exception but ignore all the others. Instead they should throw exceptions like NoPrivilegesException or DirectoriesNotWritable and the rendering / handling should be done somewhere else outside of the class that throws it. For instance in an "ErrorController" or an event or something else. I'm not sure right now what the best solution is as I'd have to think about it for a while. This is especially important for lower level classes like session and filechecks.

Note:
It is often probably not a good idea to handle it in the controller that is triggering a method. Imagine you are calling any method in any controller and then you want to catch a specific exception like this:

try {
    $model->performAction();
} catch (FileNotWriteableException $e) {
}

Now the controller needs to have the knowledge that at some point deep down there must be something file related but actually the controller should not know this. They are coupled now. Meaning whenever you change something in an underlying layer you'll have to update all controllers etc. This is something we should have in mind when looking for a solution.

It is also important that plugins can render / handle exceptions they throw and that not all error handling is done for instance in one plugin. Otherwise one plugin has the knowledge of all possible exceptions etc and they would be coupled again.