I'm sure he died, eventually....his trail lead down into the deepest darkest hell hole you can imagine....
I just had her go look, she was shooting 100 grain Remington Core-lokt
2 things, yep that is the way they die. They are usually possible to go get with enough experience, lights, time and raw determination. I'm sure you had the last 2...
I think your bullet may have been the downfall, frankly. The 243 is a pretty small fast round. Core lots hitting bone at full speed likely disinigrate. That is why I would look at bullets first. A Partition, Bonded or full copper will expand, but are tough enough to make some kind of exit hole.
So, I think she should go get her next bull with a better bullet in her 243. Likely these are $55 a box, but I guess that is cheaper than a new gun and a better fit for her. My dad quit on his 30'06 for this same reason. If I were him, I would go back to it, but with better bullets! He had to go to the spendy bullets anyways cause the 270 WSM blows them up at 150 yards!