photo or video?
Short answer: Depends. What is your iso/shutter, are you using nd filters, is it sunny or cloudy? Your exposure meter will tell you the answer.
Long answer: It depends.
For photos, I do the following. I start with my shutter at at least 1000, then set my aperture to get the desired depth of field, then adjust iso for a clean exposure. If I can, turn the shutter up to 2000+ and the iso up a little too. Most dslr's can handle iso 1600 fine, you might need a little noise reduction, but assuming you will be shooting raw and using lightroom, this may add as much as 10 seconds to your workflow. As far as f4 being too bright, I have shot at f2.8 in the sun, you just need a low iso and very high shutter (and maybe nd filters).
For video, many people like shooting with a 180* shutter (so 50 for 24p, 60 for 30p), while others like shooting at up to 250+, especially in 60p. Play around with it, see what you like, there are many examples on youtube comparing the two. Personally, I think the higher shutter looks better if you will be shooting at 60p or higher, because it keeps the subject nice and crisp when you slow it down. On the other hand, it can make some things look a little odd, especially rotating things (wheels on car, blades on heli/drone). One thing to note is that if you are shooting under electric lights, keep you shutter and framerate at a multiple of 60 (30p) in the US, or 50 (25p) in most of Europe. This is because electricity operates on the 60/50 hz frequency (respectively), and videos can look quite strange if your frame and shutter are mismatched to this. Anyways, getting back to the original question, you want to pick your shutter for the look you want, aperture for depth of field, and iso for exposure. Typically, I like to stop down to at least f8 where almost any lens (especially for video) is tack sharp, and I don't have to worry too much about focus (especially on the wide angles). If you want to have more freedom with this, say shoot shallow DoF in the daytime, you need ND (neutral density filters). These cut out the light, but evenly, think of them like sunglasses with no color cast (on the nicer ones). You can get a set of them, or you can use a variable ND filter which will allow you a continuous range (typically 2-8 stops). As far as shooting at f4, check your exposure, but I have a feeling you will need to crank the shutter to avoid over exposing (without an ND), and you will also want to check the focus, as (depending on focal length) f4 can give a pretty shallow depth of field.
This turned into quite the wall of text, let me know if you need anything explained.