I don't know how you would manage to get it stuck in the barrel. You sometimes need to put a bit of muscle into it to overcome friction, but there's nothing for it to get "stuck" on. Edit: Unless you are using the wrong sized snake...
Dan, I'm not sure you're doing anything wrong. After a few thousand rounds and/or if you shoot cast bullets, you may want to do a deeper cleaning with some stronger chemicals and a bit more scrubbing, but for regular range work the snake is 100% fine.
Much like there's no magic bullet, there's no magic solvent/tool for cleaning your gun. You can keep a firearm running just fine using a boresnake, if you do your part.