The Apollo LM first flew in earth orbit on the Apollo 9 mission in March 1969. Its first test flight was only 4 months before the Apollo 11 mission landed one on the moon! And the first all-up Saturn V flew unmanned in November 1967. The incentives and funding were there very rapid development.

Its first complete test flight, but there were years of development and testing of mockups and subsystems before.

A quick check also reveals that the LM carried in excess of 10 tons of fuel, when Dragon carries less than 1.5.

Dragon is only capable of crash-landing using its engines? I.e. there is not enough propellant, legs, etc... to do a soft landing? (not even considering take-off right now)

Depends on the rocket. With some modifications and the right rocket it should be possible to land a Dragon 2 on the moon. However, there it will be for a very long time. No chance to come back.

If you want to send up more propellant, you need to send up even more propellant to get the propellant you want up to space. Which means you have no room left for your actual payload.

Back in 2013, before SpaceX had recovered any rockets at all, Elon Musk talked about how the way they were going to get reusable rockets was by making the rocket and the reusability more efficient: https://youtu.be/vDwzmJpI4io?t=26m30s

It can land on Earth, because it uses the atmosphere to shed most of its speed, and rockets for the very last couple hundred miles an hour. No such luck on the airless Moon.

Not on the Moon. On Earth or Mars it can aerobrake away most of its orbital velocity but you can't do that on Mars. The Dragon 2 only carries 400 m/s of delta-v and you'd need 1.7 km/s to get from low lunar orbit to the surface even ignoring the need to brake into an orbit.