The Orion Project (motto: Mars by 1965, Saturn by 1970 which was killed in 1964) got around those problems completely. However, there was not only a total failure of political will, but also the concept was politically incorrect. And so we are stuck using second rate technologies and methods to this day.
“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.