The concept of a "destination paper" has always been very relative. If you're happy where you are and with what you're making and with the work you're doing, you may very well be at that destination paper. If you're married and have kids or in your 50s or 60s, you may very well be at that destination paper whether you like it or not.
As a man in his 20s, I am confident that there can be no such thing as a destination newspaper because newspapers won't exist when I reach my destination, whatever that may be.
With that said, I don't think it's a stretch to say there are many people out there who still aspire to work for The New York Times, The Washington Post, The Wall Street Journal, USA Today, Los Angeles Times, Chicago Tribune, etc. And I don't think it's any stretch to say there are people who aspire to work at Sports Illustrated, Yahoo! Sports, ESPN, CBS Sports, etc.
Your destination paper may be the biggest paper in your state. It may be the biggest in your county. It's a matter of relativity. There are still many jobs to strive for, though.
As far as any job in this industry being "relatively safe"? Nope. But you chose this line of work. Get out of it if you want a job you can stay at for the next 30 years.