is there a possibility to read information trough the serial device even if there is no boot device?
See Ingo's answer -- which may or may not provide useful information in the context of your question.
I would expect there to be some way other than connecting a screen to figure out that the device is not booting and why.
This kind of circumvents the concept of "broken", referring to something that does not work. There maybe degrees of brokenness, but at some point it must mean there is no useful direct source of information because the device is broken beyond that point. Most of the time boot failures do not mean the Pi is physically broken, but since we can speak of "broken software", that is an arbitrary perspective that again tries to befuddle the concept, broken.
95% of the time the cause is some degree of corruption on the SD card because it was not created properly, or the system was shut down uncleanly. If you can put the card in another computer that can access all of it properly (ie., not a stock Windows box) and run diagnostics on the filesystems, then that is a useful form of information.
The closest thing to what you are looking for would be the state of the green ACT led. For common problems with the contents of the SD card, this will flash in a regular pattern -- start here and read downwards to get to the detials about that.
If the green led does not do this, that page may still be of assistance. Additionally, there is the boot problems sticky maintained by the Pi Foundation discussion forum.
If the green led doesn't flash but flickers intermittently for 10+ seconds then stops (or continues indefinitely), then the OS has booted, but this does not mean it is fully functional. The best source of info in that case is going to be plugging in a screen presuming you have
/boot/config.txt. If the serial console or sshd works, then there is also that.
If the green led doesn't do anything, or comes on and stays on, try unplugging it, taking out the SD card and plugging it back in. Chances are it does exactly the same thing, meaning the card cannot be read at all properly. This can happen even with a small amount of corruption if it is in the right place. You may be able to salvage it using another computer (or at least salvage some of the data if there is anything important), but if that's not an option (it may take a certain amount of skill), then you need to reflash the card.