To add to what @AloysiusDefenestrate says above in a Comment, look at some trim-head screws. Trim-head screws have coincidentally come up here twice recently, and while some have very small heads screws with this name on the pack do vary quite a bit as to head size. The most recent mention highlighted that some have heads barely different to some normal screws.
However you could get no head at all if you wanted, but you don't want to — the hold of screws is primarily based on the head and the threading working together.
That's how screws work as clamps, the hold of the thread in the second piece and the pressure exerted by the head in the first. Take out the latter and you get no clamping force between them, so all you end up with is a fancy alignment aid1.
I know I can countersink, fill and paint over
You can do better than that. Welcome to the world of counterbores and wood plugs.
If you're sinking the screw heads below the surface such that filler is an option you're already creating a counterbore:
All you need to do is add wooden plugs to the equation.
While you can plug the holes with sections of dowel it's better to use face-grain plugs and they make plug cutters precisely for this purpose. The best kind make plugs with a slight taper, which ensure a snug fit in the counterbores.
In addition to hiding the location of the holes better (if desired) when not painting, face-grain plugs are easier to make flush (because it's not end grain that's exposed but more face grain like the surrounding wood). Also in the long term they tend to stay hidden under paint better than both end-grain plugs or filler (any type2), which tend to telegraph their location over time2 due to differences in shrinkage and expansion.
1 For that you could simply use dowels, which are easier/faster to drill for and much faster to install.
2 According to a pro I trust, his observations over his career are that all fillers eventually become not flush with the surrounding wood.