Choice of cost function in adaptive noise cancellation
You are actually asking 2 separate questions. If I have interpreted your post correctly, they are:


Why is the RLS used in Noise Cancellation, whereas the LMS is used in signal estimation?
(If we assume that the signal is ergodic) Isn't minimising the Ensemble average, $E\{e_n^2 \}$, equivalent to minimising the time average, $ \sum_{i = 1}^{N}e_i^2$; i.e. Shouldn't the LMS perform as good as the RLS?  


Answers:

Q1:

You are free to choose ANY adaptive filtering algorithm for ANY application (e.g. Adaptive Noise Cancellation -ANC).  It need not be the RLS. The same goes with signal estimation, or any other application you can think of. 

However, if you have a specific application in mind, and you know something about the system or how the algorithm performs in that setting, you may want to choose one adaptive algorithm over another. (E.g. If your application is to do with converging faster as opposed to having a low error at steady state, you might choose Algorithm X over Y) 

Once again, this choice is not made based on whether you minimize the mean squared error or the sum of the squared error. 



Q2:

This questions still keeps me up on some nights. Fundamentally, Minimum Mean Squared  Error
based algorithms are based on, finding the system parameters which minimise the $ E\{ e_n^2 \}$. Now, the expected value of the error 

Both the LMS and the RLS are based on minimising this quantity. How they do it, however, is quite different. Remember, that the Expected value is statistical operator - it's the average over all realisations of the random signal. In real life applications, we only have access to one realisation of this signal. 

Example: Let's say you have the price of a stock, all you get is one value at each time (e.g. £10) The stock price never says, "oh well I have a 5% chance of being £10, on the other hand I could be £12, or £14.....etc". But I digress.

SO, 
How are we going to minimize the mean squared error, when we can't compute the true mean? 
The Least Mean Square (Widrow-Hoff) says, "You know what, I am going to drop the $E\{.\}$ and minimise $e_n^2$. Yes it is a hack, but my god what a hack! 

The RLS uses the fact you mentioned: Since the process is ergodic, you can get the time average instead of the ensemble. The sum squared error is equivalent to the average of the squared error. The only thing that's missing is the $1/N$ term in the sum squared error. It is discarded because, it cancels out with another $1/N$ term in the derivation of the RLS. So don't worry. 

The interesting thing is how this difference affects the performance of the 2 filters. (This is a whole different question and I suggest that you ask it here - it might get a lot of people interested) The RLS is known to converge faster than the LMS. But when tracking time-varying parameters the LMS can perform better than the RLS in certain conditions. 



LMS:
$$ E\{e_n^2 \} \approx e_n^2$$
RLS:
$$ E\{e_n^2 \} \approx \frac{1}{N} \sum_{i =1}^N e_i^2$$

To sum up: Both the RLS and LMS try to approximate $E\{e^2\}$. They way they do it is different - so the way they behave is different. And you are free to choose any filter for any application.