The principled reason to oppose the draft is certainly on grounds that it constitutes a form of slavery. Not everyone is even cut out to be a soldier. Aside from the principled reasons why it is a bad idea, the military itself does not like the idea and would strongly resist it.
I think that people are pressuring the federal government in a number of areas constantly, but I think it is much more deaf than it was in the 1960s and 1970s. Street protests seem to have zero effect on policy for two decades now. On top of that, protesters get their asses kicked and are hauled off and detained. Government has responded to protests by demanding permits to organize, setting up free speech zones, and always having a zillion riot police on hand for any type of gathering. Any serious 1960s protest would be deemed illegal and it would be broken up by force.
Petitions, including the new white house petitions, are routinely talked around or ignored. In fact, the people who answer the petitions usually talk down to the petitioners, "Well, this is the policy and its for the best so we're not going to change it".
Write your congressperson? LOL. Better include an enormous bribe or else it will get a robo answer if any at all. Certainly won't change policy. Heck, there have been times in the past six years where tens of thousands of people have rung the phones off the hook at Congress. They have literally closed down all the circuits for hours and all it does is delay the inevitable. Defeat one Internet censorship bill and two more will pass a year later.
The people have done all kinds of things from across the whole political spectrum for twenty years and it makes no difference at all. If you use the parties and the electoral system, they will play dirty and change the rules. They hate us all and they look down condescending at us like children and don't care what we say. They are not public servants anymore, they have become wanna be masters.