Is this something that has happened since a change to the system or configuration was made, or has the problem always existed, from the beginning?
Are you absolutely sure that the getty accepts tty input, or is it just that input is being echoed?
Is there something like a "Single User" mode that can be used to examine system logs, etc?
If it's something that happened after a change, such as changing the baud, I'd wonder if somehow the baud rate isn't supported by the device.
But in either case, I'd hope to find some "Single User" mode, etc., that perhaps didn't prevent me from logging in, to "grep" through the system logs, to search for answers.
In some of the most mysterious cases, I've found it helpful to run something like a C program I've constructed, in place of a getty, to try get some answers.
If it's an embedded system that by default doesn't use persistent storage for logs etc., but if there is removable persistent storage available, I'd try to rebuild it so that the removable persistent storage is used for logging, so I could remove the storage and examine the logs.
If it supports a serial console mode, and you are able to make use of the serial ports at all, I might try that approach.
I hope this helps.