View LogCat from unrooted android device without computer assistance
Without root or a PC: you can't. As you noted, the behavior you're seeing is entirely intentional, and cannot be changed. If you could find a custom ROM that did not enforce these restrictions then you could feasibly install that, but if you do not want to root then I would assume this is also not an option (and I would imagine such a ROM may not exist). The only other option would be to write your own LogCat viewer within your apps, since each app can access its own log data.