how is it a dedicated video camera can accomplish this task but DSLRs struggle?
You partially answered your own question:
greater DOF would help eliminate most of the need, so maybe a lens with a good DOF would be a good idea for video regardless of camera.
Most consumer camcorders have relatively deep DoF. Although, the other part is that they are dedicated video cameras. i.e. to flip that around, one might ask: why can't dedicated video cameras capture RAW stills/photos when DSLRs have no trouble doing this? Different tools designed for different tasks.
And while shooting with deep DoF will reduce focus issues, what will you do when the conditions you're shooting under do not allow you to do this? e.g. you want to video your kid's soccer game, which is played outdoors at night.
Which leads to the answer to your overall question:
how difficult (or not) it is to keep things in focus?
It depends. How far are you from your subject? How much ambient light is there? How much movement is there? How fast is the movement? And "difficult" relative to what?