“The Orion Project”
I think that was the one that was going to use nuclear bombs as propulsion. I can see doing that in space, and it has been proposed as a means to get to Proxima Centari. But the original Orion Project had them using nukes starting on the ground. Yes, I think that would be a political problem.
I saw an interesting video where they proposed to launch a massive number of micro probes which would weigh about a gram and have light sails ~1m x 1m. They would be propelled to 0.2C by a huge array of ground based lasers. Massive redundancy, get there in 20 years. Given the advances in electronic and micro machines, I can see this tech being developed.
One problem with a single big ship is that the kinetic energy of just a speck of dust at those speeds would be like a nuke going off. Paint flecks cracked windows on the space shuttle going at ~18000 miles per hour. At 0.1C you are going 18000 miles per second. Small and redundant is possible solution to this issue.
“using current technology and methods”
That’s where basic research comes in.
Also any stationary atoms encounter at those speeds would end up with energies akin to a particle accelerator. In this case the particle isn’t be accelerated the target - the space is! If it hits moving atoms or particles just add those energies in!
It was only a political problem at the time because of the various treaties we had signed with Russian and the UN. The Limited Test Ban Treaty prevented it.
Yes, originally it was using nukes to launch, but was changed to using chemical rockets to orbit before it was canceled. One of the largest factors in cancellation was the Air Force which had other ideas like Apollo.
Collision is always a problem. However there were/are several schemes for overcoming this problem.
If you never try you never will.