Hey, let's not forget the dot on the 12th fret. This dot is here when the 12th isn't too obvious (like when a uke has more than 12 frets).
Let's answer your question with a question:
If they put dots on every fret, then would you agree that the dots then become useless?
The dot placements are selected to best help you navigate the fretboard without putting too many dots (which would cause confusion). Sometimes the dot on the 10th is moved to the 9th fret too. So their locations aren't set in stone. In fact, if you've ever played a classical guitar, you'll know that those guitars don't have any dots at all.
So, we don't want too many dots and we don't want no dots; that leaves the few dots you mentioned. Now, on to their placements. The placement must the spread out so that you can have visual markers for the entire fretboard. Ok, that means no adjacent dots. How about a dot next to the nut at frets 0 and 1? That's kind of a waste isn't it because there's already that big nut there to tell you "this is fret zero", and the "no adjacent dot" rule means fret 1 dot will be redundant.
So, we've eliminated dots on frets 0 and 1. Now, how about fret 2?
Now, a lot of ukulele stuff is taken from guitars. Maybe moving the dots around on the uke vs guitar isn't too bad an idea since the uke the tuned gCEA. But anyway, things are what they are.
So, let's see about putting a dot on fret 2. That dot would mark the notes: A, D, F#, B; looks not too bad from a C-major scale perspective. I.e. you'll be able to find all those nice diatonic notes on all the strings (and F# too, which is a note on the commonly used G-major scale). So, for the ukulele, I'd say a dot on fret #2 isn't a bad idea since we play a lot of C- and G-major songs.
But how about moving that fret #2 dot to fret #3? This has an immediate benefit of making fret #3 easy to identify, which automatically makes fret #2 easy to identify (remember our no-adjacent-dots rule). So, you get all the benefits of a dot on #2. Because of the no-adjacent-dots and no-dots-everywhere rules, to further we can push up the dots on the fretboard, the better (because we'll end up with less dots).
There you go. Let's put one on fret #3.
You can use similar argument for the rest of the dots. Except for the 12th dot. The 12 is an octave and it is a central reference mark. So, when you get close to the 12th, you can work your way down the fretboard. Therefore, many guitars put a dot on the 9th (which is the 3rd fret down from the 12th).
Oh, BTW, I just made the whole thing above up. But it makes sense to me
I'd like to heard others views.