without separating the two?
I would like to discuss this point a bit. Unlike the reality your GUI imparts, the line and your fill are separate items for the computer. They happen to reference the same path but that's all. There is no deeper reason or coupling fill and stroke than that it was easy to accomplish and seemed at one time a good option.
Now, it is by no means inferior modeling to separate the stroke from the fill entirely as separate curves. After all this convenience has some limitations. So you shouldn't couple the ideas that do not necessarily represent a fundamental principle too tightly together. If you do the danger is that your limiting your potential by including stuff in your model that are same. Do not limits yourself to other peoples modeling decisions.
add any number of gaps in any place along a stroke on an object
Now if you must have gaps in the stroke of your object, instead of having a separate object for a stroke, the yes, it is possible to do. However, I am not sure it is worth the effort.
The question is what do you gain from such a maneuver? After all clearing stroke of the fill, copying the fill, clearing stroke of that stroking that and cutting, is very straightforward easy and controllable. This is probably the way you should choose. But i will grant you that there are uses for keeping them together like needing to animate the gap. So ultimately you need to decide is it worth it.
I feel like this should be a simple technique
So here are 2 ways how you can do it, while keeping it same object!
You can use the dashed line option just specify a really long dash. You'd be somewhat working in the dark. But this is a perfectly workable solution.
The limitation is how many items you can have in the dash. You can work around this by having multiple strokes in the appearance panel but that is already really convoluted and non practical.
Image 1: Using custom dash.
Using a art brush. Wat you do is you draw a line and cut it into pieces. then save as a art brush and apply that to your object. You can make this easier by measuring the object length in the info palette and making your stroke that long. This is just somewhat pain to update. Even this has some problems other than creation.
Tip. You can add a anchor to indicate where your path starts,to control the highlight.
Image 2: Dashed stroke using art brushes.
In the end is a question of whether the work is worth it and what you gain. Also is it simple? Well that's for you to decide.