A thousand thanks for that @jessegrabowski , it helped understand a bit more.
Although I still have even more questions 
By actually plotting the log-likelihood i get the same shape as you posted above. I had already done it in non log space and I see a world of difference in the two.
Here is contour plot of the likelihood (sorry about the scale on x axis (p)) (a)
and here is the corresponding log-likelihood (b):
-
So, by looking at (a) it seems that the likelihood has a local maximum at approx. the true values of p,n. And indeed the model seems to home in on that when n_experiments is somewhere ~ 1000.
-
Looking at (b) i now agree with what @jessegrabowski posted above, but how can we explain 1) in this context, is it just random that the true values seem to be found in this case ? (I have not yet really experimented with changing N=20 and P=0.5 as shown in the first post)
BR