Short Stop, The first thing you need to mention when describing any product is its version. In the case you bring up, 4K maybe a version issue, but since I do not have any 4K monitors I cannot say. But I have been using Snag-it for at least a decade, and full screen feature has always been available. I have Snag-it Versions 11 and 12, so if you are complaining about Version 13 or the new 2018, your problem may be an issue of configuring the product. This was the easiest to do with Version 11, and became much more complicated and LESS clear with Version 12. I can only suspect that 13 and 2018 are even harder to setup. In version 12 and prior there is a setting that allows you to select the capture field to be a Region, or the entire screen. It sounds like you have Region set. I have Full HD monitors (1920x1080) and there is no issue capturing any portion or all of the screen.
When using Region, you should be able to start the screen selection by placing the cursor in the top left corner of your screen, and then slide it down (holding the mouse button down) all the way to the right bottom corner. If you can do that, once you let go of the mouse button, it should capture the whole screen. This works the same for a single image capture as well as for video capture. Again, the newer version have a terrible GUI (User interface) and setting it up is much more complicated.
Paul - Snag-It works quite well with video capturing, but I only use this feature when the other methods do not work. Real Player Download works with most sites, and captures both the video and audio nicely, but only when it works! Some sites block this. So, then I use Snag-it to capture the video along with the audio. But when using Snag-it to capture the video, it makes the computer almost unusable for other things, because the frame selected is fixed, and any window that pops up onto it will also be captured!
In my opinion, Version 11 was the best! I have not purchased any since Version 12, and use Version 11 99% of the time!