What is the reason why Buddhist monks don't work to support themselves? Do the monastic rules prohibit them from working? It may be hard to find a comparative monastic tradition in the west although there are some monastic in Christianity currently most are really assimilated and intertwine with the Church which bears its existence in charity; so it's not practical to compare these with Buddhist monastic tradition. However, if we go back in history there were monastics in Christianity completely detached from the church, for example, the desert fathers ( Christian ascetics in Egypt and Syria ) were completely detached from the church, but  following the Apostel's command that 'a person unwilling to work should not eat' they lived by the labour of their hand.

So, the question, in short, is why do Buddhist monks rely on alms for sustenance? 

I don't mean to sound sarcastic, but for many observers, it's perplexing to see a young and strong monk asking for sustenance from old and ragged and gray women.
I recommend you to read the Kasi Bharadvaja Sutta. Kasi Bharadvaja was a farmer who asked this exact question from the Buddha. Monks dedicate their lives to spiritual development. They do not have time to make money or do agriculture. They do not eat for sport or beautification. They only eat enough to maintain life so that they can work towards Nibbana. Yes, farming does break a precept.

"Should any bhikkhu dig soil or have it dug, it is [an offence of Confession.]"