I suspect there are two reasons why they won't tell us their expected completion date. The first one is that there is no system in place to predict the completion date. There could be, and it would be simple. Something like "there are 56 applications ahead of you, and on average the last 100 took 0.7 days each, so your will be done in about 56*0.7=39 working days". But there is no such system, so you don't get updates. The second reason is they don't want to tell you a date and then miss it. Even though you would much rather have some kind of estimate, even a slightly inaccurate one, companies and governments in general don't like to commit to deadlines unless they are forced to.
Edit: Actually, thinking about it now, it's probably about fairness. Even though they could give a reasonably accurate estimate to 90% of applicants, the other 10% (complicated applications) would be really hard to predict due to individual circumstantces. If they give estimates to the 90%, they have to give them to the 10% too. This would mean the estimates for the 10% are either totally wrong (way too short, because they would be based on averages that include simple applications), or they would require significant coordination with multiple other departments to come up with a realistic estimate. So they just don't give estimates to anyone.
