Personally, I wouldn't know what the 'correct' cause of action would be if it were the government, but something would need to be done. You are probably totally correct that airstrikes wouldn't achieve anything.
My view is simply this: If I was living in Syria and my government were launching chemical weapons, I would pretty much accept anyones help - and considering the US, UK and France are some of the biggest powers, it's our duty to step in when something's wrong. As I said before though, a large part of this is dependent on who is launching the attacks.
As for your last point, who knows what will happen afterwards. But surely (if it were the government), Assad being gone is a better thing.