I had similar issues with cities becoming road blocks between mountain passes. With the inability to select exactly where garrisoned troops appear when leaving cities I basically became trapped in one tiny corner of the map until I demolished everything in the city.

Note sure what the reasoning behind not being able to walk on city improvement tiles was, but it does not make for fun gameplay.

when you put your unit in the city, you can select their profile icon and then right-click to move them out of the city where you'd like.  Of course, it would be nice if they would pathfind through the city.

I had one game where I built a city right next to that gems deposit by the southeastern lake. Couldn't send my pioneer through the city to get it. He couldn't even leave it after stationing in the city. Similar stuff happens when there's no way around a city. You have to garrison, and then let them out the other side. It seems, too, that units don't leave a city right now to it's borders, but one or two squares away. Hence why that nice tile was unaccessible.