Why do Air Forces still use bombers?
Once you launch an ICBM, you're pretty much committed.  Even if there is a self-destruct on the missile (I don't know, nor am I making that claim.  Additionally, I think debris from such an act would be a big concern.).  A bomber also allows you to send something part of the way, and rattle your sabers in the face of the other party without again having committed to the act of the warhead being launched.  You can always recall a bomber without having to worry about a debris field or other unintended consequences. 

As to the claim that bombers are particularly vulnerable, there are numerous ways they get around this.  For instance the B-2 employs stealth to get around this.  The B-52 can use standoff weapons to reach out to a target (such as cruise missiles or JASM).  Also, the B-52 can use low altitude routes to avoid radar and detection (if you’ve never seen a B-52 do a low level, I recommend you find a route and observe, it’s something else).  And on top of all that, the aircraft have defensive systems to help them on their missions (chaff, flares, RW gear, etc.).

Also, with a bomber, you can employ different types of weapons that give you a broader range of response options.  You can use bombs of anywhere from a 250lb laser/GPS guided bomb to something much larger.  These delivery mechanisms are generally much more accurate than a TBM (Theatre Ballistic Missile) or even an ICBM (InterContinental Ballistic Missile).  Those systems are generally inertially guided (some have GPS capability), which can lead to a larger CEP (circular error of probability), meaning that it may not hit exactly where you aim.  Those systems rely on a much larger warhead to get the job done than bombs from a bomber.  You don't use a Nuke to swat a mosquito, as a bit of military wisdom goes.  And, even ballistic missiles can be intercepted by systems such as the Patriot, so they are not a panacea against being stopped.