You've stumbled upon a dread secret of
permaculture, Dan. We don't like to talk about it, but the truth is there is no proper way to end a swale; a true swale must continue indefinitely until it reconnects to itself, following the contour line of course. If the contour line doesn't behave this way,, or leaves your property, too bad, you can't use swales.
...thankfully no. There are a few options that come to mind, and you're right to be concerned about it... but I think you're already on the right track. The key is really the low spot which will be either the planned or de-facto spillway, and this certainly doesn't need to be at the end of the swale. To close the end just angle the berm or swale/berm upslope as you described, and then put a deliberate spillway wherever the water will be most useful/least inconvenient.
Some of the so-called 'swales' in my area are slightly altered by addition of a very, very gentle slope, to help move water across the landscape to a more desired area gradually, rather than it all soaking in until reaching overflow level.
Breaking your swales up into sections has some merit in that as you describe, in case of a washout less water will escape. On the other hand, you'll either need more spillways for each section, or more likely make your swale-dividers lower than swale level, so that overflow occurs first between the segments rather than over the swale itself. You would also loose some of the spreading effect of the swale in light rains, if that's an issue. I would probably choose to put the effort into more, bigger swales, and robust spillways, rather than dividing up my swales...
As far as your tractor access, roadways of any sort can really flow a lot of water; you might give some thought, while planning your swales, to modifying this access route so that it will dump any flowing water into swales rather than having it all continue down the access way.
If there is perceived high risk of swale overflow *and* this is an undesirable outcome as it is not practical to channel the water somewhere productive or at least harmless, ideally one would add more swales, or deepen the swales, or use other methods like ditches or drains to get the flow under control, until such an overflow became extremely improbable.