most of my left hand dumbek technique is just fingertips on the edge (or that sweet spot just inside the edge).  like you said "just dropping a finger on the head while the wrist remains on the side of the drum"  sometimes ill roll my first three fingers (my pinky is pretty useless   )  i get most of my variety from my right hand and use the left to fill gaps and extend finger rolls that i start on my right hand.  sometimes ill lift my wrist to get a lower tone from the center of the head but rarely.  

ive got two dumbeks (or a dumbek and darabouka)  one is a turkish style with about an inch of rim and a 10.5 inch head.  my left hand can really only do the edge tap stuff on that drum.  my other is a smaller drum and my left hand fingers can reach in a little more to the center.

ive also been known to mount 'em on my kit which totally frees up both hands and i can play it like a djembe and get full range out of each hand.

Ok, I am TEKing with my ring finger only and it sound nice.  Problem is, I can't do it fast enough and my finger gets tired and weak.  

the pattern that is troubling me is

(1)DOUM   (ee)     (&)TEK    (ah)TEK  (2) repeat
(both left handed TEKS)

It's the two quick back to back TEKs that don't sound clear if I have the metronome higher that 100bpm.  Does this come with time?  Or do I need to hit the weights?  I am a flamenco guitarist and feel that my finger strength isn't the issue, .....it must be my technique.  

funny, id be more prone to alternate ... doum right, first tek left, second tek right.  seems it would give more time for your right hand to switch between the two ...

but thats just me.

ahhh ... my bad, i was reading his (2) wrong ... which of course means you are spot on right.   ill shut my yap now  

I have some of suggestions.

If you're using "traditional" position........

Use a belt.  The drum will be more secure this way, so you won't have to worry about it falling off your leg.  Just tie it around your body, and your doumbek.  Something wide/and or padded is desired here, so you won't have too much pressure on sensitive/vital organ spots.

Use a stool or guitarist's leg rest on the leg that holds the drum up.  Now, you can bring your ka hand in from the lower side of the drum, rather than holding over the top.  This will be easier on the muscles in that arm.

I have a chronic problem with my left shoulder, and playing traditional position aggravated this.  So, now I'm playing my doumbek on a stand, which is angled away from me, like the position of a djembe, or the snare position of a very old school jazz drummer.  There are two advantages.  

For one, you don't have to hold the drum up.  The stand will do that for you.

Also, both hands are now equal.  The ka hand is no longer the weak partner, and you can easily play doums with it.

LP makes an adjustable stand that will  fit a doumbek, if you want to try this method.