I am glad you detected that the actuarial student is correct. This was an obvious point many students missed out on.
$N$ will typically be a very large number, which in turn makes $E_x^C$ very large. Now, since your Poisson probability mass function for the Poisson model contains an exponential term, values greater than $N$ will be VERY small.
Now, does this mean that we ignore the fact that the Poisson can give probabilities for more deaths than what was actually observed in your sample? I have two answers to this (either I would think is okay)
1. Yes, you can say that this is a flaw of the model, and you can just ignore these deaths. Although typically, the number of deaths will be VERY SMALL compared to the total population, so we will not really run into these situations in practice. However, what may happen at very extreme ages, such as age 100, where your death data and population data is roughly equal? Think about this (for extension purposes).
2. Maybe not.... What if we want to apply our Poisson model to another situation, where the population in that situation is greater than the population you estimated the parameters on?
I hope I have also answered your last question. Another point to think about, in comparing the models, is practically/realistically speaking, do we really use them at higher ages, or mainly for ages where we have more data (and hence better statistical reliability)? Think about this as well.