What makes a bomber able to deliver nuclear bombs?
You are right in saying that's all that's needed, with the addition of a strategic or tactical need.  

The need for a "special evasion" technique is only needed for aircraft dropping megaton range weapons in an air burst.  Kiloton range weapons could be dropped from altitude with no special evasion required except a lot of speed.  Sub-sonic aircraft would need to be "frisky" in their escape. Read about Tsar Bomba for a good example which demonstrates that even with the biggest bomb ever dropped, a relatively slow, lumbering aircraft can escape.

Even helicopters can drop nukes.

Most aircraft are only tasked to drop tactical, low yield weapons.  The real madness is assigned to subs and missiles.

As the comment above says, I suspect that most crews dropping strategic weapons will not care much whether they survive or not.